I'll learn the notes for a cover song, but play them with my own "feel". I've just learnt I Want You Back by the Jackson 5. The original bass part is fairly stacatto, but I play it with a bit more "flow" between the notes. As long as what you're doing suits the song, I don't think it matters if your own personality comes through.
If it's for a tribute band though, I think it'd have to be spot on to the original as much as possible.