Tough choice. But I'll say Alice. Mostly because their music is more popular here and I didn't have much contact with Soundgarden. I have only seen that video with smiling people and heard a couple more songs, but they sound good too. Sorry if my choice isn't backed up with more objective reasoning.
2 really fine bands. I would have to say Sound Garden because I really like Chris Cornell. The best thing about Alice in chains is that they can play so amny styles of Grunge. They have a unique sound.
BTW, godsmack (one of my top favorite bands) sarted as the offical Alice in Chain Tribute band. Check them out
I, being a HUGE grunge fan, this was a very tough choice. Alice In Chains is overall the better band, with amazing guitar riffs and great choruses. However, Soundgarden has some amazing hooks, and Chris Cornell is one of the most amazing voices in all of rock music. I'd have to go with Alice In Chains, because of their influnce on so many people.