Take, for instance, the trusty Sniper class. Its role in TF2 is the same as it was in TFC. You have a big gun, you point it at things, and those things die. I was good at this; if there were a Last Starfighter Sniper, I'd at least have made bootcamp. But now, a lot of the tools I'd come to rely upon are gone. First off, I don't have any grenades (no one does) to protect myself from the faster guys who can get close to me and zig-zag around, avoiding my powerful shot. My backup weapon, which used to be the sniper rifle on full-auto, is instead a crappy SMG. And the worst shock was even my sniper rifle had changed. I used to be able to kill all but the strongest classes with a good chest shot, and now I typically can't bring down another sniper unless it's a direct headshot. I quickly realized, if I wanted to be a good Sniper now, I'd need to - gasp - rely on my team to keep me safe.
And that's a new feeling for me, a TFC veteran. Back then, teams were a loose confederation of people wearing similar clothing who agreed not to shoot at one another, and when a naive server admin would turn friendly fire on, guys like me would name themselves "GoodTeammate" and ignore that agreement as it suited them. Everyone had his own objective - some people liked chasing down flags, others like racking up kill totals - and if they happened to overlap, it could occasionally look like they were working together.
TF2 really alters and improves that chemistry. Most obvious is the team score, which is displayed prominently in the lower center of your screen; a friendly reminder that everyone should at least pretend to want to capture the flag or control some specific territory. Funnily enough, it works. The people who care only about their score typically don't stick around long in close games, because their teammates are too busy yelling at them to participate with everyone else. And the game actually rewards teamwork, regardless. Scores now, which used to be enemy kill tallies, are a lot more representative of someone's contribution. For example, you get points for helping someone kill an enemy, for knocking out an opposing sentry turret, or for defending an objective the enemy is trying to capture. The game makes you want to be a part of a functioning team, rather than just being the best player wearing red.
Most of the differences between the two games are to nudge you in that teamwork direction. Take the Medic class. In TFC, they could heal friendly players and infect enemies with a contagious plague that gradually killed the other team. In TF2, they instead heal other players, which builds up an "ubercharge" meter gradually. When the ubercharge meter is full, the Medic can activate it and render a teammate and himself invincible for a little while. It's a great strategy for breaking up turtled enemies, and it's also an excuse for at least two people on the same team to coordinate. The aforementioned removal of grenades works the same way. You can't just go solo into the enemy base and hope to blast your way through. Now, you need to organize an actual assault to get past the enemy's defenses.
And the game is much, much better for it. People actually use team chat to accomplish things, rather than just yell racial epithets at one another. Compare this to Halo 2, with its rather unruly clientele on Xbox Live, and I can't help but wonder if it's just a lack of direction that turns 12-year-olds into idiots.