In the United States, we use both. That's really all you can say about it. "Fall" is slightly more common, "autumn" is slightly more formal. When we are naming the seasons in order, we normally say "fall," as in "spring, summer, fall, winter." By the way, in current US usage, we don't capitalize the names of the seasons. I've seen them capitalized in old books (e.g. from the 1800s).

"Fall" is chiefly North American.