Graphics are far less important than the gameplay mechanics. Story can be irrelevant in some games though... I'll point to the Twisted Metal series as a glaring example of this. What was the story? I don't think anyone knows honestly. Battle royal with supernatural vehicles, but it's fun! I'd also point to first person shooter games which don't require much story either.
One game where the story was extremely lacking was Diablo 3. Not only was the story lacking, but the entire feel of the campaigns was weak compared to Diablo 2.
Diablo 2 was dark and macabre, but Diablo 3 felt like an animated title from a major studio like Dreamworks or Disney. It was kind of campy and terrible.
Warcraft 3 (also from Blizzard) did not rely on story really. Sure the campaign did, but most of the fun was in the Battle.net player vs. player just like Starcraft.
Diablo 3's campaign and online games were sort of one in the same. The online play was simply the campaign with multiple players.