Base-Breaking Character: Scouts. Some think he's a great offensive class, but others regard him to be useless when compared to the medic.
The Combat Medic is one of the most popular classes to play, being a great offensive and supportive class, but there are people who think he's overpowered when it comes to his many abilities.
Is the Pyro hard but enjoyable to play, or the most useless class in the game and just a mere gimmick after her nerf from TF1 to TFC?
Depending on the person, grenades are either a nice feature that is necessary to use, or incredibly stupid and only useful to spam with.
The radically different sequel is also a big controversy with a lot of TFC players either enjoying it or hating it completely. The TF2's community mostly hatred of TFC isn't helping either.
However, the massive spike in popularity from the TFC team's surprise appearance in the TF2 webcomic as the 1930s team, the fanbase demanding more classic-inspired items/maps in the game (just look at the workshop) and the Original pretty much being the primary for choice for many, many, soldiers, both communities are actually getting along quite well.
Crowning Moment of Funny: The Smeer The Queer community map. The goal of the map is to have someone on one of four teams wear a Teletubbies costume so his team can score points over time. The map has a children art style, that combined with the game's overall seriousness, can be quite funny to play for the first time.
Demonic Spiders: With enough skill, any class (except maybe Scouts) can becomes this, but here are some specific examples.
Combat Engineers are really a nuisance if you are playing offense vs defense. Building teleporters in your base with usually a sentry guarding it while they go around killing your teammates by the 3's with their EMP grenades.
Infect and run Medics can be a REAL pain when your team has no medics to heal your teammates, while you and a few others are chasing the speedy fucker as you are losing your health by the second.
Face of the Band: Although not as much as in TF2, the Heavy Weapons Guy was the only class that was on the box art and the only one that is on the main menu.
The Team Fortress 2 webcomic solidifies this by having the Classic Heavy being shown as the leader of the retired squad.
First Installment Wins: Some players still prefer the original Team Fortress for its fun and frantic gameplay, and it's simple but more competitive feel over the second game.
Memetic Mutation: "Enjoy your AIDS!" and "EXCUSE ME! I'M IN NEED OF MEDICAL ATTENTION!"
Most Annoying Sound: "Medic!" calls tend to be a ear-grating even for non-Medics, especially if you're near someone who won't stop spamming it.
Most Wonderful Sound: Hearing the SPLAT! sound after gibbing someone is very satisfying, and it lets the other player know how hard he/she got his/her ass kicked.
The sound effect after getting a backpack or medkit.
Also getting healed by a medic and getting your armor repaired by an engineer.
Older Than They Think: Most people believe that the teleporter was added for the first time in TFC. They were in the first Team Fortress for only one version after it was removed for being a game breaker (there was no wait for the teleporter to recharge, unlike in the later games).
Also in the early versions of Team Fortress, the scout had the double barrel shotgun in his arsenal, but was removed for presumably being too cheap and it wasn't until TF2 that he got it back as an exclusive weapon.
Paranoia Fuel: Any time you are around a corner, you could find yourself face to face with any one of the other classes (or a sentry) out to kill you. At least you can hear them... most of the time.
The only way to know if that grenade thrown next to you was your or the other team is whether or not your ass is 5 yards away from your head after it explodes.
See that red dot on the front of your base, that the enemy sniper, waiting for someone to come out of the base...
Is the corpse of your team's soldier really a corpse, or an enemy spy waiting for someone to pass by...
Sequel Displacement: Team Fortress Classic was this to the original Quake mod (go ahead, ask anyone if they're even aware that Classic is not the first Team Fortress). Team Fortress 2 wound up doing this to Classic itself.
Team Fortress 2's influence can also be felt within the game. Spies and Demomen are now more common, and Medics actually heal people. Before, only a small handful used them, and only went around infecting people with a virus, using the speed of the class.
"Stop Having Fun" Guys: People will bitch about grenades, call people noobs for not using grenades, and get angry at TF2 for not having grenades. In fact, there's no fighting over classes - just grenades, and maybe Sniper Rifles and the virus attack the medic has.
That One Level: The Rock (or rock2) can be a real pain to play with the gimmick of having to stop fighting and find a gas mask or a place to avoid the poison gas that covers the entire map and removes a point from your score if you happen to die from it.
Tier-Induced Scrappy: The Heavy is the easiest class to play, and having the fat bastard kill you the 20th time without a problem or chance to fight back can be a real annoyance to some players.
It's not uncommon to hear people call the Medic top-tier, with his high-speed Concussion Grenade jumping, pretty good damage output at medium range, and deadly infection from his medkit melee weapon.
With her Nerf from the change from TF1 to TFC, the Pyro is regarded the most useless class with her slow and weak flamethrower and the Medic and Soldier is always picked over her.