I'd like to agree with several other opinions here, that I can't separate art from writing, and that both have always gone hand-in-hand for me (or at least drawing and making up stories-- I can't say I was really writing much when I was 4), and I would have a near-impossible time quitting either.
But I can't honestly say they carry equal weight in my sense of identity. And I didn't know that I valued one over the other (as far as which one I 'identify' with more) until I got a comment from a close family member that I was inexplicably really offended by: My dad, probably twenty years ago by now, once told me he always thought that my writing was better than my art, and that if I ever 'made it', it would probably be from something I wrote, rather than something I drew.
... And for some reason, I had never been so offended about something someone had said about my art. And I realized that if the statement had been reversed, and he'd thought my art was better than my writing? I don't think I'd have been offended by that at all.
So that kind of showed me that, at my core, I identify more as 'artist' than 'writer', even though I do both.