I believe AmE uses just 'movie'; BE uses both interchangeably. The word 'movie' is more recent, so trendy and popular with young people; 'movie' also tended to be used (regardless of fashion) by critics/students of film.
Some collocations are more likely with one word than the other; Google's not working for me at the moment, but I imagine, in UK pages, 'movie buff' is more common than 'film buff', and 'that movie-going public" is more common than 'film-going public'; but I'm not aware of any difference in semantics.
This interchangeability is quite recent. Within my life-time, 'movie' was much the less common of the two; in the 1950s people would go to see a film, or even a 'flick'.