Another point in favor of tampon applicators: it is possible to wet them, which on occasion is practically a necessity. You aren't always gushing blood whilst inserting a tampon; neither are you generally particularly aroused. Maybe you are trickling blood, but maybe not, especially by the third or fourth day of your period. As well, you often need to insert a tampon immediately upon removing a used one, and I find that tampons tend to, you know, ABSORB things, leaving me devoid of much of the normal liquid in that bit of my system. So, natural wetness is not always enough lubrication for insertion of this strange little cylinder of compressed superabsorbent cotton.
The natural and convenient solution to uncomfortable, dry insertion is water. However, there is a slight problem with wetting plain tampons. See, tampons are fibrous, and meant to expand in order to absorb blood. But they can absorb other liquids as well! And when they absorb liquids, they get less firm and potentially misshapen. It can be like trying to insert a wet straw wrapper. Obviously, this is a bit of an exaggeration (no one wants to pre-soak the thing), but it's not very appealing, and not all that successful either. You want the tampon in its proper shape for proper insertion.
An applicator, in contrast, is made of rigid cardboard, sometimes coated with plastic, or just of plastic. It is smooth and not particularly absorbent. It is, of course, harder to work with plain cardboard, but certainly easier than just the tampon. You can wet it without getting the tampon inside more than very slightly damp, if at all. To do this, you can hold the applicator under a very thin stream of tap water, avoiding the very tip, or lick the outside of it (an excellent solution in a public restroom stall). Licking, incidentally, is also a point against the non-applicator tampon: although I have not tried this, I would bet that millions of little cotton fibers would stick to your tongue were you to lick a plain tampon. You can lick a piece of cardboard, however, with no problem.
So, after you wet the applicator, you have your necessary lubrication. You can then insert the tampon without undue pain or irritation.
Lucy-S points out that lube such as K-Y would also work. I don't generally have any lube on me while out in public, personally, but perhaps you do. Anyway, yes, this would probably eliminate the need for an applicator.
fuzzy and blue says: "if you're using tampons that turn too mushy to insert with a little water, they're also likely to leave fibers in you and increase your risk of TSS. you might want to switch brands." Well, I didn't know that, so in case you didn't either, here you go.