Yes, they're the same season — autumn is simply the British name, and fall is the American one. Fall started life as a shortening of "the fall of the leaf," so the two words are close cousins in origin as well as meaning.
The core difference
Autumn and fall both mean the season between summer and winter; autumn is preferred in British English and fall in American English.
- autumn — the season between summer and winter: The trees turn gold and red every autumn.
- fall — the same season, standard in American English; also a verb meaning to drop downward: We always visit New England in the fall to see the leaves change.
How to tell them apart
Reach for autumn in British English and fall in American English; either is understood on both sides of the Atlantic, but each dialect has a clear preference for its own term.
Fall does far more work than autumn, though. Beyond the season, it's the everyday verb for dropping down (leaves fall from the trees), for decreasing (prices fell sharply), and even for a government or city being defeated (the city fell after a long siege). Autumn stays put as a noun for the season, and stretches only as far as describing a decline, as in "the autumn of his career."