Take for example SWTOR. I played through the game the first time with a Trooper. What did I enjoy the most? The stories. Combat was just a way to get myself to the next bit of story. I didn't care about gear, PvP, crafting... just the story. My second play through was on a Jedi Knight. I really enjoyed seeing my Jedi's story arc unfold. And I really started to hate running around and fighting. Eventually, I just got so tired of the fighting that I didn't even care if I saw the rest of my story arc. The same was true for every character class in the game (I had one character for each so that I could play through each story.) The burden of combat eventually became such a weight that I just left the game. When I let my subscription lapse, I had only played through my trooper's arc to the end, 3/4 of my Jedi Knight's arc, about 1/4 of the Smuggler arc, and a few levels worth of my Sith Warrior. The other classes, I left almost completely untouched. (I played through the first part of the Bounty Hunter in beta, but never got around to replaying it after the game went live.)
After six months, I was pretty much done with SWTOR. CoH kept me around for 8+ years.
It's all the little things that CoH did to break the 'stock' MMO playstyle that kept it attractive. I've been playing SWTOR since the CoH servers went dark, and if what you want is to experience a different storyline, there's no real reason to ever create more than eight characters -- one for each base class from each faction. Each base class, regardless of advanced class, has exactly the same class missions, and there are so few non-class missions that you're pretty much stuck with doing all of them on each character. CoH, even at the end with every new character fed into the same 'starter' mission series that introduced them to gameplay, could toss it all as soon as they got introduced to other contacts, or even ignore it from the start and street sweep or do sewer trials, and there was never any
one storyline -- you could select the contacts you wanted for each new introduction, and pick which story arcs you wanted to complete. While the general sweep of advancement might be similar, I would be surprised if any two characters completed exactly the same missions on their way from 1 to 50.
And other MMOs seem to be mired in both the "3 minions = 1 hero" and 'role trinity' concepts that CoH threw out the window -- SWTOR, for example, explicitly has you select from 'Tank', 'DPS', and 'Heal' as your role when using the group finder to queue for flashpoints, and it's programmed to assemble groups as 1 tank, 2 DPS, and 1 heal. There's none of the "Who wants to do [TF]?" and taking anyone regardless of class or build that CoH had.
I enjoy SWTOR, but it's not the "get lost in playing a character and find out that I've been at it for six hours" fun that I had with CoH; too often, it slides from fun to tedium after only an hour or so, and I need to take a break and read or play a different game for a while.