Both of them mean "very" according to many dictionaries, but if I remember it correctly, sometimes きわめて and いたって are interchangeable, sometimes not. How to use them correctly?
きわめて is just a stilted way of saying "very" or "highly".
According to the results of this corpus of modern Japanese, I can say いたって 1) is less common overall, 2) often describes either the personality of a person or the perceived atmosphere of a place, and 3) tend to have positive or reassuring meanings.
For example, いたって普通 sounds slightly more positive than きわめて普通. Both means "normal", but the former can mean "working normally" or "no problem", whereas the latter is closer to "unremarkable" or "undistinguished".
至って難しい and 至って不健康だ sound odd to me. According to this list, however, 至って seems to have been used more freely 100 years ago.