I can't explain WHY it is so, but if I say I am going to see a movie, I mean that I am going to leave the house and go to a cinema. If I say I am going to watch a movie, I mean that I am going to stay home and look at the movie on my TV.

it's the same

the difference goes to US and UK pronunciation and phrases, see in UK and watch in US are getting used normally.

I agree with Hilary. There isn't a difference in meaning between 'see a movie' and 'watch a movie', but there is a difference in usage. As native speakers, we mainly use 'watch' for films/ movies at home/on TV, and 'see' for when we go out to the cinema (GB/Aus) / movie theater (US). We also use 'see' and 'watch' in the same way as we use these verbs in other contexts. 'See' is a sense or a chance event, whereas 'watch' is a more conscious act. So you would say, for example 'I was watching a movie when I suddenly felt ill', because in this sentence you are talking about the activity you were engaged in.