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.