It is a synonym to programming-challenge, but they are both after the exact same thing. No differences. Except a programming challenge sounds better than a contest problem, because that makes it seem like there's an issue with a contest!
I believe this can be changed. Contest problems are still considered challenges. The only difference is that contest problems may have certain criteria. If that's the case, then they should be stated in the question. We don't get too many actual contest problems, and even then, they're usually not approached too differently from regular programming challenges.
Instead of burninating the tag, I'd recommend making it a synonym. They are quite similar, and it will still show that the question came from a competition.