“Flautist” is a pet peeve of mine. I just encountered it again in a message board thread.
To summarize: there doesn’t seem to be any good reason for English-speaking people to say “flautist.” “Flutist” makes much more sense. Most dictionaries list both, but I’d venture to say that’s only because “flautist” has become acceptable through common usage (if enough people say it wrong, it’s right!).
I’ve heard people say now and then, though I can’t find it documented anywhere, that the terms are gender specific. For example, “flutist” refers to a male and “flautist” refers to a female. I’ve also heard it the other way around. I don’t know why we would have gender-specific terms for flute players when we don’t have them for other instruments.
For what it’s worth, I’ve also observed that “flautist” seems to be especially popular among non-musicians; musicians, level-headed bunch that they are, seem more likely to say “flutist.”