This is a spin-off thread from Ras's thread about covering a 60s tune.
I'm with those who don't mind changes in cover versions. I often substitute chords (like an Am for a Cmaj), do the tune in a different key, mess with the melody lines, add harmonies, etc. Certainly, I get creative with arrangements, in some cases because I can't play the instruments used in the original
I tend to think trying to sound like the original isn't a good strategy, because you'll always not be what people remember and love.
Also, some songs work in several different ways, so it's fun to put on a "different set of clothes" on the song and see it from a different perspective. For example, I'm doing a cover version of "Magie Noire," which to me is sort of the perfect mid-80s French pop electronic dance music. No way I could do it better. But, I'm doing a more guitar-centric treatment, and giving the vocals less urgency...in the original, the singer resisted trying to be brought into the girl's spell, in my version I'm more resigned to it as "the way it is." So it's the same song...different treatment, and instead of people comparing it to the original (e.g., "that sucks"), they'll hopefully see the song in a different way, and appreciate it all the more.
But live...I dunno. I think a lot of cover bands sort of have to sound like the original, or face angry crowds throwing beer bottles at them. When I do covers live, which is rare, I do them very differently and people don't complain. Then again, they tend to be more concert situations rather than "house band" environments