The arcade version of Contra appears to be very similar to the home version at first glance. The first level of each game is almost identical, while the second through fourth levels introduce some changes, and after that it is very different.
Konami originally designed Contra as an arcade conversion kit. That meant a few things. The first of which meant that it got a vertical screen, because vertical conversion kits sold better than horizontal ones due the the huge installed base of vertical games from the early 80s (today pretty much everything is horizontal).
The arcade version looked and sounded a lot nicer than the NES version. That had a lot to do with the fact that it had more than twice the processing power, and a nifty YM2151 audio chip. Unfortunately that is the only department it beats the NES version in. It loses in every other comparison. I will get into the specifics of that below.
First off, it was just a lot easier than the NES version. I was able to beat it using 13 lives on my very first try (I can't even beat NES Contra on 13 lives, and I have been playing that version for 15 years now). The bosses in particular go down quite a bit faster.
Secondly, it is a much shorter game than the NES version. The NES version has 8 levels, while the arcade version only had 5. The first level in each is nearly identical. The tunnel levels are a bit different in each version (NES version is longer and harder), and the waterfall level is shorter in the arcade and has a different boss.
Then we move onto the 5th level. The 5th level in the arcade is pretty long (longest level in the game), but it is also the last level in the game. The NES version takes elements from this level and uses them to create levels 5 through 8. The final boss is the same, but it takes almost twice as many hits to destroy on the NES version.
Why all the changes? It is quite simple really. The arcade version was too short and too easy to port directly to the NES. Everyone would have it mastered in a day. It didn't matter that it was easy at the arcade, because it only took 10 or 12 minutes to beat the entire game (at which point it is over). So they lengthened the game considerably and beefed up the difficulty in the home version, which is what created a classic NES game out of an arcade "also ran."