Nearly everyone who plays flute will, at some point, be asked "What is the correct term for a flute player, Flutist or Flautist?" The answer is; either one. Yes really. Flutist is the older English word with the Oxford dictionary dating it to 1603. Flautist is a much more recent term in spite of the persistent rumor to the contrary. Nathanial Hawthorne seems to have been the first writer to use Flautist ("The flautist poured his breath in quick puffs of jollity" from The Marble Faun) in 1860. Hawthorne may have been trying to sound European (a popular fad in his day) by inventing a word based on the Italian flauto (from old Occitan flaut) to get the word Flautist. Older English writers (like Chaucer) use floute/Floutour, flowte/Flowtour and my favorite, floyte/floytynge (playing the flute) with no connection to the Italian term. Floute/Floutour, flowte/Flowtour and floyte all developed into flute/Flutist in English. Dictionaries currently list both Flutist and Flautist as correct terms for flute players.
Many people think Flutist is more common in the USA but in my experience, both get used about equally. Common musicians' gossip says that Flautist is more common in England (or Europe) but again, that doesn't seem to be true in practice; in fact, in my (highly limited) experience, Flautist is used more often in the USA than in England. Some people feel Flautist is stuffy or even somewhat insulting; they claim its too similar to "flaunting" or "flouting" in spite of not being based on, or related to, either word. On the other side of the debate, some people feel Flutist is an Americanization or less technically correct; both ideas are also completely untrue. It strikes me that the objections to both words are more based on emotional reactions to (and personal associations with) the words themselves rather than on dictionary meanings of the words or their true history.
The reality is words change over time and usage shifts back and forth for mysterious reasons (take a linguistics class if you want to really hurt your head with this phenomenon.) The great Flutist vs. Flautist debate has been going on almost since the word Flautist was tossed into English and it hasn't changed much in all that time. I prefer to skip the whole issue and say "I am a Flute Player" or "I play the Flute". But I answer to Flutist, Flautist and even Fluter without complaint or regret.
Just make sure you know me before you call me a Flutter brain.