For me, the importance of story depends on what type of game I am playing and how the gameplay fares.

Example: I could care less about the story in a fighting game, odd stuff like Katamari Damacy/Rez, and puzzle games (if a story mode is available, of course). I am just playing to have fun.

Example 2: If the gameplay is really fun but the story is lacking (see: FFX-2), then I can still enjoy myself. However, if the gameplay sucks but the story is awesome (see: 24 the game), then I can also enjoy myself.