Tunes you wrote and recorded
This is what we're all hoping for. You've got something to say and you want to take it to the world. Go for it! This pretty much depends on having some song-writing talent, BTW.
Tunes you recorded but didn't write
At one time, writing your own material was almost unheard of (although it seems to have worked well enough from the Jurassic era right up until big recording companies appeared). Writers wrote and players played. Fact is, not all of us are song writers, and not all song-writers are performers.
Cover version, original concept
When someone takes another crack at a previously recorded tune, the results will sometimes over-shadow the original: "Blinded By The Light" (Manfred Mann vs. Bruce Springsteen), "All Along The Watchtower" (Jimmi Hendrix vs. Bob Dylan). A different artist can often add a lot of value to an existing tune, often adapting it to a different genre. And when well done it's a thing of beauty.
The music business relies on recorded tunes that someone else wrote. It's a vast and ever-growing catalogue (check out the Billboard 100 from time to time). Most musicians get their music from recordings. If they didn't, the music business might just grind to a halt. It's value added for artist(s), keeping their music alive, and it's what a lot of audiences are in the mood to hear.
And in conclusion
All musicians play cover tunes … it's where the majority of music comes from. And all bands are cover bands, at least in the beginning. Artists depend on the tunes and the tunes depend on other artists. But sadly, the term cover seems to have morphed into almost an insult. Nobody is offended when they remake a movie, such as Godzilla (not once but twice!) And no way would anyone protest, “It’s just a cover of the Ishiro Honda movie” or suggest that the film-maker had cheated or copped out.
In the end, a great tune is a great tune, and a great rendition is a great rendition, no matter where the tune came from. If you're in a so-called cover band, you are part of the foundation of the music industry, and you should be proud to be a member of this thriving musical community.