Yes and no. Language is never quite that simple, and the US/GB distinction is never as straightforward as people think.
American English uses both words, but most commonly 'fall'. AE speakers say 'fall' for everyday use, and 'autumn' to sound romantic or poetic. It's typical, I believe, for American English speakers to talk about the beautiful 'autumn colours' that you can see around this time of year. So, even though 'fall' is the word which AE speakers use most of the time, this does not make 'autumn' a specifically British word. For Americans, it's just a less common alternative.
In British English, we always say 'autumn'. Nobody says 'fall' in Britain. The only exception is the little mnemonic which Norman mentions above, and British children who haven't picked up 'fall' from American TV might need to have 'fall backward' explained to them. That said, most people are aware that 'fall' is used in American English.