To start, CO uses the Champions setting, by Hero Games. Champions is your standard super-hero setting, focused around Millenium City (formerly known as Detroit, thank Doctor Destroyer's version of 'urban renewal' for the name change). And while it may not be anything groundbreaking in terms of superhero settings, it has something that CoH didn't have: an established mythos to build off of.
Let's be honest here, the whole storytelling aspect of CoH and CoV tended to be formulaic and repetitive, which is what happens when you're building a world from scratch. With CO, however, they were able to build on an established universe, with decades of plots woven into the setting. What does this mean for the person who's never played in a Champions game? It means that, unlike CoH, you get characters who have a lot more depth to them, right off the bat, such as the villainous (and deranged) Foxbat, who uses a custom made gun that fires ping-pong balls. Ordinary, unmodified ping-pong balls. I kid you not.
Beyond the mythos of the setting, there are some vast improvements over CoH when it comes to the game itself, starting with character generation (chargen, for short).
CO takes the basic costume designer from CoH, and improves upon it, adding in more options, and more ways to customize your character's build. Even better, gone are the days where you had to wait to level 20 to get a cape, or play a certain amount of time to get a trenchcoat, or craft wings to put on your costume. There are costume pieces you can unlock via different items and achievements in game, but the big things, such as being able to play the cloaked hero, or the person with angel wings is open to you right at the start.
Then comes the 'crunch' of chargen, picking powers and abilities. In CoH, you pick an archetype, pick one of a few 'themes', and you had your starting powers assigned to you. If you saw one Cold type blaster, you've seen them all. Not so in CO. Starting right at Chargen, you can dispense with archetypes, and mix and match powers based on what you want to do. Perhaps you want your energy building attack to be using twin swords, and your special attack to be a lightning bolt? Done. You want to make that lightning bolt red, and have it come out of your chest? You can do that, too.
One of the other major problems with CoH was that they gave you movement powers like you got your mount in World of Warcraft. At level 7, you could get a basic power that allowed you to slowly hover in place. It wasn't until grinding through to level 14 that you actually got to fly. And if you wanted to do an ice slide, or rocket boots, or an energy disk? Yeah, not so much.
In CO, you get a travel power once you leave the tutorial area, and you aren't limited to just Leaping, Super-speed, Flight, and the like. Tunneling? We got that. Teleporting? We got that, too. And this isn't a crap power that'll have you eating a tortoise's dust. This is an actual movement power, allowing you to fully explore the zones.
Ah, the zones. You know how CoH had mobsters on every street corner, accosting people and stealing purses, even when they were strong enough to beat the crap out of rookie heroes (if they ever ventured out of Perez to go to Atlas), and you'd just go around beating them up until you got what you needed? Sure, there are mobsters around Millennium City, and yes, you do occasionally see them menacing civilians, but its no where near as bad as you see in CoH.
Missions. There are still missions, of course. You've got three basic types: manhunt, collection, and escort. Manhunts are when you go around a certain section of the city (or, more rarely, in a building) and hunt a person, or group of people. Collection missions are when you find a certain type of item (either lying about in containers, or as loot from defeated enemies). Escort missions are the bane of your existence. You typically end up following along behind a person who has little fighting skill, and if you die, or they die, or you get too far apart, or you get jumped by a patrol, you fail, and have to start over. And the people never walk faster than a slow stroll.
Still, gone are the days of nearly identical building maps with goons from different gangs in almost the exact same positions. For the most part, you don't go into the buildings, and when you do, each one I've seen so far has been a unique layout, with different challenges.
I could go on, but then there'd be nothing for you to discover in the game. If you're looking for a new game to play, pick up Champions Online, for either the X-box or the PC. You won't regret it.