As a drummer, I think it is very important to tune the snare as well as the kick drum and the toms.
Let the song dictate what the snare needs. The pitch has everything to do with the feel of the song. Your video at http://www.youtube.com/watch?v=behK0UE2Eoc shows that pretty well, but you could really make it better since your library has much greater potential.
Just an added note: As far as the playing goes, if you listen to a lot of the old 60's recordings from Stax (Sam & Dave, Booker T. & The M.G.'s, Eddie Floyd) you will notice a famous technique drummer Al Jackson discovered. He slightly delayed the snare notes to give the song a kind of shuffling feel, like the music wanted to speed up but the drum kept it in line.