My approach to playing is the same in both my originals and covers band.
I have a song structure to follow and a bass line to play which I can embellish here and there with fills etc but MY opinion is that my role as a bass player is to support the song/band. In my originals band I try and write a bass part which at best enhances the song or at least fits the song, a skill I learned from playing covers and listening to my fellow band mates as well as recording of other bands/players. I've played my own songs more in the last 6 months than anyone other tunes so I'm basically covering myself at every practice and gig!
Cutting to the chase my point is with playing live, as long as the punters (and the band are enjoying it), it doesn't matter who wrote the song in the first place as you'll likely have already played it 100's of times be it a cover or original. I enjoy the money from gigging covers and the reception from an appreciative crowd, especially if it's our own material. I wouldn't give up my originals band to play covers exclusively or vice versa and fortunately I don't have to