Characters and Story
I've never played the original Final Fantasy III (I understand it was only ever officially released in Japan) but I recently borrowed a copy of the Nintendo DS remake. Apparently there are a few minor changes to the story, the job system has been changed, and the graphics have been massively updated with 3D characters and new, higher resolution sprites for the enemies. The characters are no longer called Onion Kids, instead going by Freelancers, although there is a hidden gimmicky job you can get called the Onion Knight.
The characters are all given personalities for the DS remake but they don't affect gameplay at all. The game starts off with Luneth falling into a hole in a cave, needing to fight his way back out. Soon after he meets Arc, who leads him to a quest to save the people of a neighboring village who have been turned into ghosts. Along the way they pick up Refia, the spunky daughter of a blacksmith, and Ingus, a taciturn soldier in the service of the local king. After saving the village from the curse, the characters discover they have been chosen by the four elemental crystals as their champions to restore the balance of light and darkness to the world.
The characters are all rendered as polygonal 3D models that look like toddlers wearing armor. This gives them an impressive degree of freedom to act out cutscenes, swing weapons, and walk around fully animated while appearing in the clothing of their current job. Dungeons, towns, and buildings are all 3D as well, with a fixed camera angle following the character. You can even zoom in, which sometimes reveals secret switches and hidden treasures.
Rather than having a fixed class system for the party, like Final Fantasy I and II, and most other computer role-playing games, at major turning points in the story, the characters are granted the option of training in new jobs. This continues throughout the story, allowing the characters access to more potentially powerful jobs as the story advances. There are certain points in the game where it's strongly recommended by the plot to have a certain job in the party to make a dungeon easier or defeat a certain boss character.
Because of the job system, characters have experience levels and job levels they must advance. As far as I can tell, experience levels only really affect how many hit points your character has. Everything else appears to be a function of the current job level, from attributes to magic points. All the characters start out as Freelancers, a very weak job (beating the game as Freelancers is an additional challenge some fans of the game like to try), and should pick a new job as soon as they're offered.
Job levels are not increased by defeating monsters, but by taking actions. The capacity points system in the original has been discarded in favor of a new system. Any action taken during a battle, such as swinging a weapon or casting a spell (even guarding, but not moving to the front or back row or running away) counts to advance the job level. After about 5-8 actions or so (depending on the current job and level) the character increases a job level. All jobs start out at level 1, but changing to a new job doesn't erase the progress made in the old one, the character can return to the job at any time and continue from where he left off. Jobs can be changed at any time, but changing to a new job requires a period of adjustment, during which the character's attributes are lower. A character's current job and job level determine his attributes, magic points, the type of magic he can learn, and the types of armor and weapons he can wear.
A cheap, although tedious, way of quickly increasing job levels is to go to an area with weak enemies and just choose the "guard" option instead of attacking. Since guarding counts as an action, doing this five or so times before killing the enemy will increase a job level at the end of the battle.
Freelancer - Jack of all trades, master of none, Freelancers can use some weak magic and the early armor and weapons in the game but their usefulness runs out quickly. Change to a new job as soon as you can, unless you want to beat the game with the Freelancer challenge (this is apparently very difficult).
Special Ability: Low level white and black magic.
Warrior - Has the ability to use most swords and heavy armor. The Warrior is very oriented to offensive capability rather than defense. Recommended.
Special Ability: Advance (damage bonus but sacrifices some defense if attacked)
Monk - A bare-handed brawler, the monk does high damage with his bare fists or he can equip claws, but cannot use heavy armor. Not so good.
Special Ability: Retaliate (will counterattack the enemy for massive damage, but will not attack if he is not attacked that round).
White Mage - Uses weak weapons and armor but can cast powerful healing spells to keep the party alive. Also casts spells like Toad and Mini to get the party into hard-to-reach places and can reverse some status effects. The white magic in FFIII seems a bit overpowered though, very often he will have several castings of cure spells available that are powerful enough to almost completely restore the entire party in one casting, making the party invulnerable to anything that can't kill a character in one hit. Since this is a role-playing game, you pretty much need a dedicated healer in your party. Recommended until the very end of the game, when you might want to switch to the Devout.
Special Ability: All but the highest level white magic.
Black Mage - Uses weak weapons and armor but can cast powerful offensive magic in battle. The Black Mage's spells are most useful when used against enemies weak to certain damage types, for example using lightning spells on water-type monsters. Unfortunately the Black Mage is oveshadowed by the Geomancer, Evoker, and Summoner later in the game. Not useful long-term.
Special Ability: All but the highest level black magic.
Red Mage - Like a Freelancer on steroids, the Red Mage is a jack of all trades but is actually pretty good at all of them. He can use a lot of powerful weapons and armor and low- to mid-level white and black magic, making him the most versatile character in the game. Recommended early on, but you'll probably want to change to a better class late in the game.
Special Ability: Low- to mid-level white and black magic.
Thief - The Thief uses knives and boomerangs and some halfway decent armor, but has two special abilities: unlocking doors without keys and stealing from enemies. Unfortunately both of these abilities, and the Thief in general, are pretty useless. Keys are cheap enough that unlocking doors isn't a hassle and 9 out of 10 times all he's going to steal is a cure potion. There is one unique weapon in the game that can only be stolen, but it's not worth the trouble. Not so good.
Special Ability: Unlocking doors and stealing.
Ranger - The Ranger uses bows and decent armor to attack from the back row for extra protection while still doing full damage. Arrows are cheap enough that filling up on them isn't much of a drain on your purse, but he's essentially a Warrior you have to pay for. Not recommended.
Special Ability: Barrage (fires multiple arrows at multiple targets).
Knight - If the Warrior focuses on offense at the expense of defense, the Knight is the opposite. Late in the game the Knight gets some impressive weapons though to make up for this. The Knight can also use level 1 white magic that is good for refreshing your party between battles, letting your White Mage save his spells for the boss fight. But his most impressive ability is to automatically step in front of characters who are down to critical levels of hit points, taking the hit for them so they don't die (although this doesn't work for magic attacks). Unfortunately, many of the more powerful boss attacks can still kill a character who isn't down to critical hit point levels so it's not as good as it could be. Recommended.
Special Ability: Defend and level 1 white magic. The Knight has a "Defend" battle command but it doesn't appear to do anything, the Knight will protect weakened party members whether this is selected or not.
Scholar - The Scholar fights with magic books that do elemental damage, and is sort of like a Monk in most respects. The elemental damage almost makes up for the fact that there are only a few weapons he can use, so he's not upgrading his arsenal as the game progresses like the other characters are. The one thing a Scholar excels at though is using items, which are extra-powerful when used by him. Not so good.
Special Ability: Study (reveals a monster's hit points and elemental weakness). However there is a white magic spell that does the same thing.
Geomancer - The Geomancer uses elemental magic based on the environment he's currently in. There is a tradeoff here though, although he doesn't use magic points, his spells are somewhat unpredictable. He may use a damaging spell, a status effect spell, or an attempt at an instant-kill spell at random. Of the options, the damaging spell is the only one actually worth anything, since status effect and instant-kill spells rarely work on regular monsters and almost never work on bosses. When he does use a damaging spell it's generally more powerful than black magic but his unpredictable nature limits his usefulness. Not so good.
Special Ability: Terrain (random, but free, magic).
Dragoon - The Dragoon is halfway between a Warrior and Knight, equally good at offense and defense. He uses heavy armor and spears, which do wind-type damage (good against flying enemies). His special ability is Jump, which takes him completely out of the battle for a round. In the next round he falls out of the sky onto the enemy doing extra damage. Unfortunately, the Jump ability has significant drawbacks which limit his usefulness. Being out of battle for a round means he can't be targeted by attacks or magic from enemies or friends while he's away, making him difficult to heal during battle. And it also means he's not there using his heavy armor to take hits for the magic users. Not so good.
Special Ability: Jump (leaves combat for a round to do extra damage in the next round).
Viking - The Viking can use very powerful armor and weapons that do lightning-type damage. At high job levels he is very powerful but it takes a while to get him to that point. Mostly defensive in nature. Not so good.
Special Ability: Provoke (attracts enemy attacks to himself, sparing the weaker classes in the party).
Dark Knight - The Dark Knight can use many of the swords and armor in the game, plus the dark swords and dark armor. They're really only useful at one specific point in the game, though, a dungeon where ordinary attacks cause the monsters to divide into more monsters. Dark swords and magic don't cause this, so you're "supposed" to use a Dark Knight during this dungeon, but I just ran away from every battle and avoided the issue entirely. Otherwise this is pretty much a gimmick class. Not so good.
Special Ability: Souleater (sacrifices 20% of the Dark Knight's hit points to do heavy damage to the enemy party).
Evoker - The Evoker is another magic user, with the low-damage weapons and armor that implies. He uses summoning magic, which is different from white and black magic. His summons can be unpredictable, but not as unpredictable as the Geomancer's. Most of them have a 50% chance of either doing damage or a status effect spell, with two exceptions (one summon does damage or a little more damage, and one does damage or heals your party). While his spells are more powerful than a Black Mage's, the unpredictability reduces his effectiveness. You can never count on that healing spell when you really need it, for example. Not so good.
Special Ability: Summoning magic.
Bard - Ha ha ha ha... Bards.
Special Ability: Sing (mostly group-effect status effects and heal spells)
These jobs are mostly upgraded versions of other jobs you already have access to. They are more potentially powerful, but switching to them means starting your training over at level 1.
Black Belt - An upgraded version of the Monk job.
Special Ability: Boost (stores attack power to do damage in a later turn, two Boosts can be stacked for even more damage, but a third attempt will blow up in the Black Belt's face doing a lot of damage to him).
Devout - An upgraded version of the White Mage, the Devout has access to the final and most powerful level of white magic spells.
Special Ability: All white magic.
Magus - An upgraded version of the Black Mage, the Magus has access to the final and most powerful level of black magic spells. Personally, I think Geomancers, Evokers, and Summoners have better, if sometimes less predictable, magic. Not so good.
Special Ability: All black magic.
Summoner - Across the board a better version of the Evoker, the Summoner uses all the same summons but they each have only one effect, making them predictable. His summons are generally more powerful than black magic and they all affect the entire enemy party, unlike the Evoker who only affects one enemy at a time. Recommended.
Special Ability: Summoning magic.
Sage - The Sage can use any kind of magic in the game; white, black, or Evoker-style summoning (I understand in the original version, he used Summoner-type summoning). Keep in mind though that you can only learn so many spells from each level, so one Sage is far less flexible than two other magic users. Recommended only if he's your only magic user, otherwise you're better off with a Devout and something else.
Ninja - You wouldn't expect it, but the Ninja is the upgraded version of the Dark Knight. He's not so useful against run-of-the-mill enemies, but he just tears bosses apart due to his special ability. Throwing an item rather than swinging it does lots of damage, but that item is lost forever. Not so good.
Special Ability: Throw (does lots of damage but loses the thrown item).
Onion Knight - The Onion Knight is a secret class that can be unlocked by a side-quest I never found. They can use just about any weapon, armor, or spell in the game but they have lousy attributes until they hit job level 95. Once they get up to level 95-99, a process which takes hours and hours of playing, they become the most powerful and flexible job in the game. This is a gimmick class.
Special Ability: All magic.
Personally I used the Red Mage, Monk, White Mage, and Black Mage at first, but wound up using the Black Mage's bow more often than his magic. Later I switched to Red Mage, Scholar, White Mage, and Geomancer. After that the Red Mage stopped being so useful and the Geomancer was just too unpredictable, so I switched to Knight, Scholar, White Mage, and Evoker. I stayed with this party until the last dungeon when I changed to Knight, Black Belt, Devout, and Summoner, which I beat the game with. From time to time I would experiment with the other jobs but was left feeling underwhelmed with them.
Problems with the game
There isn't too much grinding in this game, but as an RPG there's going to be some by default. The most grinding is in leveling up your jobs when you change classes. Overall this game suffers from problems typical to the Final Fantasy series, however.
White magic is overpowered. It's far too easy to keep the party healed during battle and the White Mage and Devout have plenty of magic points to keep casting throughout the dungeon and boss fight. Black magic on the other hand is underpowered, there's no reason to use it anymore once you have access to the Geomancer, Evoker, or especially the Summoner. Additionally, early in the game you get magic staves that cast ice, fire, and lightning without spending magic points. Once you've got these you barely have to use your actual spells anymore because they're so powerful.
Due to the improved graphics, the game seemed to be stretching the capabilities of the Nintendo DS. To save processing power, the enemy party is limited to a maximum of 3 enemies at a time. This further decreases the usefulness of black magic, which is best used taking out large numbers of weak enemies.
The job system is silly. By changing your clothes you lose all the abilities you spent half the game training yourself to use. I would prefer a job system that allows you to level up your magic use for a while, and then level up your weapon use for a period while your magic abilities remain static.
Although sitting in the back row offers some measure of protection by increasing a character's defense, neither sitting in the back row nor being low in the party list seems to affect how often a character gets attacked. This reduces the effectiveness of meat shield characters like the Knight. It also makes melee fighters with weak armor like the Monk and Scholar (who should be in the front row to do full damage) more vulnerable than they should be.
Status effects and instant-kill spells are pretty much useless. They have such a low chance of working even against normal enemies that they're not worth the trouble of using. And you can forget about using them on bosses at all. Status effects are best done as a side-effect of another spell or weapon strike, such as the poison dagger, rather than as a dedicated attack that is wasted if it doesn't work.
There's also a serious lack of variety. Most battles are best won by hitting "attack" over and over again until the enemy dies, and there are no mini-games or puzzles to break up the monotony of wandering from dungeon to dungeon killing things. This is, however, a remake of an early RPG, when this sort of thing was less common, but modern players are used to having them and one or two could probably have been added.
The game also has little replay value, so I'd recommend borrowing or renting it rather than buying it. Many of the job classes are either gimmicks or slight variations of other job classes, so while in theory you might want to try going through the game again with a different party, the appeal really isn't there in practice.
Fun to play through once.
An excellent walkthrough by arkfullofsorrow hosted on Gamefaqs filled me in on much of what I didn't figure out myself playing through the game. Some of the differences between the DS remake and the original version were taken from Wikipedia, arkfullofsorrow's walkthrough, and Shinma's E2 writeup.