Fortune and Spirits, you may have heard of us (ha!). As a band we play a few cover songs, but on the whole, our live show is mostly original music. Which is how I prefer it. In part because I like to have people hear the songs that I write (ego), but also because I want the music we're creating to have as much face time as possible (strategy). In the long run the band, as entertainers, will have more loyalty if our fans are fans of our music.
After that you're on your own. I'm no expert so I could be totally wrong about anything (or everything). As we all know, I don't really know anything about anything ;), so let me know just how wrong I am in the comments *wink*.
I know many artists have truly MADE a song their own; Aretha Franklin comes to mind. And let me make it clear: I am a huge fan of doing covers adapted to your own style. It's possibly my favorite part of Weeds too :). But Aretha franklin didn't write the vast majority (or any?) of her own songs, which is a very different scenario from a band whose raison d'etre is writing music.