I played a rogue through to level 70 and tried my hand at several other classes. I also tried dozens of other MMOs when I got tired of WoW. Seems I just don't like MMOs that much. But there were parts I did like, and Shamus nailed all of them.
I've seen lots of other MMOs best WoW on a few of these features: City of Heroes and Villians had amazing character and play-style customization; Lord of the Rings had both great graphics and a strong crafting system; Dungeons and Dragons and Warhammer both had great combat systems; Warhammer's RvR and scenarios were fun, exciting, and rewarding.
Unfortunately, as Shamus said, they all focused on a few things and screwed up on the others. Dungeons and Dragons and Lord of the Rings had terrible social interactions -- it was difficult to find people, make groups, and stay in contact with friends. Warhammer's PvE was generally lack-luster, as was its storyline: it was basically just an MMO version of Counterstrike. Warhammer's player customization also left a lot to be desired; they said it was because everybody was supposed to be part of the same army, but I call it a cop-out -- users want customization, canon be damned. It also had the fastest mob repopulation rate I've ever seen, to the point where I regularly died from the same boss that I just finished killing because it respawned before I could leave the room. City of Heroes supposedly had a good storyline, but I never saw it, and the worlds seemed sparsely populated and repetitive. Age of Conan... well, I don't know that it did anything right.
I would actually still play WoW, but there were a few things they could have done better that new games could take advantage of:
1. Faster progression. Most of what people who don't like MMOs complain about is the slow grind to get anywhere, and I agree that they have a point. The first 10-20 levels give you levels, items, and new skills every few hours, but at higher levels it can be days or weeks before anything new and useful is gained. There's only so long you can beat on the same monsters before it gets boring; new stories, areas, monsters, items, and skills need to come often enough to keep things fresh and new, even if that means the player goes through the game's content faster.
2. More emphasis on skill and teamwork than grind and gear. Many people say that WoW doesn't start until the level cap and they're right. Until then, every encounter can be soloed, every monster can be beat with minimal thought, and it's only a matter of grinding long enough until the next level comes. But in the end game, raids demand much more of the player; groups of 5, 10, or 25 people need to work together, plan strategies, balance abilities, and otherwise work hard to make it through an instance. They are also much more rewarding, giving players items that make a difference to their abilities, often unlocking the ability to take on bigger and more difficult instances. This is what makes a game a game: challenge and reward. This shouldn't be left until the end game; this should happen all the way along.
3. Better crafting and customization. WoW allows players to have a lot of control over how their character plays through skill customization, enchanting, gem socketing, glyphs, and a host of other facilities, but where they fall down is in crafting. Most things that can be crafted aren't worth crafting by the time you acquire the ability to do so, and the ones that are useful are only useful to the player to make once. A craft is a creative art, where a person makes something better than usual through their own care and creation. If everybody can make the same stuff you can, it's not a craft, it's a commodity, and to maintain balance, it has to be valued as such, which is what happened in WoW. They managed to fix that somewhat in Wrath of the Lich King, where a player can only have a few of the best recipes, resulting in everybody having a specialty or two that few others have.
For one, crafted items should be worth crafting. They should either be useful to the person crafting it, or should be worth enough to another player that they will pay enough to make it worth it to the crafter. Crafted items should also be more flexible: let the player choose colors, designs, and attributes. There are lots of ways to keep this balanced; for example, give each potential crafted item a point value based on the power of the attributes the crafter wants it to have, then base the material cost, time to craft, and likelihood of success on that point value and the crafter's skill. By allowing the crafter to come up with the combinations, you allow them to make the item they want without having to make hundreds of recipes that are all very similar, and you allow them to express their creativity, which is every craftsman's pride.
Discovery is also fun. A lot of people like to experiment, and crafts based on chemistry, biology, and magic lend themselves nicely to this. Let people try combining random ingredients to see what happens. Sure, somebody will eventually make a website listing all the combinations, but most players never see those sites anyway, just like most players don't read the spoilers to find out the storyline.
4. Better communication. WoW is one of the better ones for this, but it took them a long time to get it even half-ways right. If you are using text chat, it needs to be clearly visible, with enough lines to maintain the context of the conversation, especially when things get heated. It needs to be clear who is speaking, including separating guilds, private messages, game messages, and global chatter. Filtering is also important. Ignore lists are essential and shouldn't be limited in their size.
Better yet, through it all away and use voice chat. For groups working together, nothing is more immediate and personal than voice and it frees the user's hands to play the game. In games without built-in voice chat (all but WoW, which only picked it up within the last couple of years), the best groups will make their own voice chat with programs like Ventrilo and TeamSpeak, but it's inconvenient to be switching between programs. You know your users want it, so build it in, and make it seamless, automatic, and expected.
There are probably other things they could use, but these are the ones that strike me as the most important. If I could find an MMO with the quality of World of Warcraft, but with these improvements, I wouldn't care what theme it was or what company developed it; I would be helplessly lost in the game.