It is both, of course. But my impression is that, over the years, it has become increasingly less a game and more a story.
For those who are not familiar with WoW, player characters have a level, increased by gaining experience—killing monsters, doing quests. The level has a maximum. Each time Blizzard does a major revision of the game—the latest is the fourth—the maximum level is raised and players who were at the old maximum set out to level up to the new.
In the original game, as best I remember it, leveling was fairly hard—you were doing quests that at least sometimes stretched your abilities, requiring ingenuity and/or patience to complete. In at least the last two versions, that was no longer the case. Almost all of the quests were easy, with the result that a competent player's character would almost never get killed—and death in WoW is not permanent, merely a minor inconvenience.
The player was being walked through an elaborate (and, in the most recent expansion, gorgeously illustrated) plot. Only when he got to the top level did it become necessary to do more difficult dungeons and raids in order to win better gear to equip the character with. In effect, the game has been split into two halves, with everything up to top level primarily a story. The top level offers the opportunity for those who wish to do relatively hard things to do them, provided they do them in groups. So far as I can tell, the only opportunities for someone playing solo to face real challenges come either from trying to skill up very fast, which can involve fighting lots of enemies at once instead of one at a time, or going solo into dungeons intended for lower level groups.
Aside from the lack of high level solo content, I'm not sure the current structure of the game doesn't make sense. It lets the designers focus their attempts to create difficult and interesting content on the top level, which is where most serious players end up spending most of their time. On the other hand, it also means that getting to that top level doesn't stretch the player and so may not teach him what he will need when he gets there.
My most serious criticism, however, is not of the structure but the content. Blizzard's designers do a fine job as artists, even if their tastes in armor and weapons are a lot farther in the comic book direction and farther from realism than mine. They do a good job as programmers.
But if any of them writes a novel, I don't think I want to read it.