Story = The narrative your character experiences, either mandated by developers or created by you Lore = Background information about the game world that you do not personally experience but which gives flavor to your story
I do not believe that story should be important compared to lore, yet developers seem to disagree with me. From voiceovers to heavily instanced story missions etc, MMORPGs increasingly resemble single player games with their emphasis on "story-driven gameplay."
I'm going through vanilla WoW again, and there is a story there, but it's completely optional and only matters if you want to pay attention. You can toggle slow quest text for example, but you can also have it appear instantly and not care. There is, however, tons of awesome lore that is all pervading.
I think at the heart of the MMO should be multiplayer gameplay and lore (not story). When I was younger, I was rather antisocial and played MMOs as if they were solo games, but they aren't. The experience is vastly more rewarding when you are social and encouraged to be social. I think that the focus on story takes away from the potential for player-created stories with friends.