The Soldier
Soldiers are the backbone of your team. They are the only class that can use the Rocket Launcher (a modified version of the Half-Life RPG, which acts like the Quake1 Rocket Launcher). Along with the rocket launcher, the Soldier can also carry 200 red armor and 100 health, but at the cost of a greatly reduced movement speed. Soldiers also carry the Nail Grenade, which will hover in the air when thrown and spit out nails in every direction until they explode like a normal grenade. Soldiers are probably the most versatile class because the rocket launcher is a weapon that can be used in almost any situation. Soldiers will be seen in every facet of TFC game play, from offense to defense and everything in between. Their primary drawback remains the slow movement speed.
What's New?
The biggest different I saw with the soldier is that you load one rocket at a time (visibly you see your hand putting rockets into the launcher) and at any point during the reload process you can stop reloading and fire a rocket, given there is at least one rocket loaded into the launcher. In other words, if you shoot all four of your rockets, you can fire again as soon as ONE rocket is in the clip. You don't have to wait for all 4 to load like you used to in TF1. You can also switch weapons while you are reloading. Good? Bad? I'm still up in the air on this one. Right now it's just a fact, but it's a huge change.

This soldier won't be hurting anyone for a while.
Rockets have a faster rate of fire in TFC; the trade off is that they travel slower. Also the rockets have a bright white smoke trail on the end of them, and are readily visible because of this. This flare and the white trail they leave behind them easily give away the position of the soldier that fired it: no more rockets from out of nowhere!
The jumping animations from Half-Life remain in TFC. That means when you jump in the air you make this obtuse gesture with your whole body. This is a lot different from Quake1, where the only thing that would happen is that your body would fly up in the air. This is a big difference because now when someone jumps, their body squishes down, decreasing the bounding box of what you can hit. It will be a lot harder to have midair shots, and it's also hard to predict their trajectory so you can shoot at the spot where they are going to land.
The fuse timing for grenades is longer for TFC than Quake1, so it takes a while to get used to pre-priming a grenade and having it go off in midair. It feels like you're waiting forever, but after playing for a while I got used to it.
Nail Grenades are slightly different for the original as well. In TF1 the nails would be shot out in a spiraling pattern, so you could time them and avoid them with a little effort. In TFC the nails are shot out in random directions, and there are many more of them, and in a Half-Life theme, the blast radius of the exploding nail grenade is much larger.
Rocket jumping still works, and the vertical height you can attain is increased over TF1. It could be the map, but you can now RJ from the ground all the way up to the attic in the map "Well." I don't mean the catwalk around the sides, but the actual ground, the same level as the respawns. Unfortunately the code for RJing is still buggy. Sometimes when I would RJ it would feel like someone was grabbing me and violently shaking me in midair. This happens a LOT when you hit the sky "ceiling" on the map 2forts.
There is no instant weapon switching!!!!! (This applies for all classes, but I take it personally for the soldier) ACK!!!! I HATE that. Probably because I'm odd, and I like to switch between my SSG and RL quickly in TF1 (you can hit a scout and bounce him in the air, and kill him with a SSG blast. Also, at point blank hitting someone with an SSG instead of a rocket saves your health). In TFC, I think it takes something like 600-800 ms to switch weapons (similar to Half-Life).
Since the water code for HL is different than Quake1, you can no longer RJ out of the water in 2forts back on to the land.