Since the two are - perhaps unfairly - linked as rock-and-roll martyrs, both being "grunge" (though AiC drew more on metal and Nirvana on punk):
I fully expect Cobain to win, owing to the size of his fanbase, but...
... in my opinion, Staley by far. To begin with, Staley could actually sing. Even at the height of his productive heroin addiction, in 1996, Staley was able to take part in the creation of the single best "Unplugged" album in MTV historiy, "Alice In Chains Unplugged":
http://www.youtube.com/watch?v=0dVz2D1xS-0
His rendition of "Down In A Hole" in particular is just heartwrenching, and even his fuck-up on "Sludge Factory" is endearing.
Cobain on the other hand, was a better guitarist. He was a worse lyricist and singer. I also feel that Staley was one of the few authentic voices in the 'grunge' scene. Vedder could emote, and Cornell could wail, but Staley genuinely felt the pain he sang about. And he died because of it.