I think the fact that this thread contains 18+ pages of passionate replies really shows how special CoH was! Here are some of the things that made CoH feel unique to me:
Roleplay
This was the first time I actually tried roleplaying in an MMO, and boy was it great: Pocket D was amazing, and it still is in Homecoming! Roleplaying really felt like a game-within-a-game, to the point where it was all I'd do some nights. Add to this a bunch of awesome super groups and bases where we could hang out, and it was incredibly immersive.
Unique Roles
I absolutely love controllers (and dominators). They provide an experience that I haven't been able to recapture in other MMOs. You really felt like you could control the battlefield, with so many ways to disable enemies. I think this role has long since been forgotten in modern MMOs in the name of "balance" (when's the last time you saw a class that could completely disable groups of enemies for more than a few seconds?). Other support roles also felt amazing. Buffs and debuffs feel so meaningful in this game. They're not just tiny stat boosts you can grant for a short period of time. With a team of support characters, you can turn of zeroes into a team of heroes!
Creativity
People were (and still are) incredibly creative with their choice of power sets, costumes, and character names. It makes teaming up together so much more fun! Every character felt unique, and their costumes weren't constrained by gear, race, or class choice (unlike many other MMOs). You could have a stylish character at level 1, whereas in other MMOs, style usually was more of an endgame thing.
Chaos
This has been mentioned before, but there's really nothing quite like running a bunch of radio missions and things like it. It's a lot of chaotic fun! I can't think of another MMO where you constantly fight crazy amounts of enemies. Bodies flying everywhere, big explosions, hurricanes, snow storms, the whole shebang. Some of the most fun I've had have been in teams where there isn't a tank, chaos erupts, and we're all fighting for our lives. Suddenly we get ambushed and it turns into a full out brawl. Ahhh! Because revival is incredibly easy in this game, no one's afraid to get their hands dirty, so this can lead to some crazy (fun) scenarios.