Quests are those plot devices or events that send you off to do an errand- kill so many of this, collect so many of this, find this, do that- without little involvement or change on the NPC's part. An example of a quest is the barrow outside Ivarstead. You're told to investigate the barrow, and that's that. Nothing really happens as a result of your completing the request.
Stories are those plot devices or events that involve the NPC in the story personally. They make you feel invested in events or characters, or elaborate on an NPC's personality, backstory or life. An example of a story is the plotline involving the Redguard barmaid in Whiterun. Depending on what decisions you make, things can turn out vastly differently, and the NPCs involved are fleshed out.
Which do you prefer, and how do you think Skyrim's handled this balance?