They are more or less the same. The only difference is if you are talking about seeing a movie at the cinema, you would normally use 'saw'. If you mean at home, or on tv, then either is ok, but it can sometimes depend on the context. The main thing to remember is for cinema, use 'see'.

You can also say you went to a show/concert/performance/movie. That's the activity. Similarly "watching" is an activity. "See" is one of those short words in English which has come to have a hundred meanings. Here it is used as a substitute for other words to take away the focus from the activity. So if someone asks you what you did last night, you could say "I went to the cinema to watch a movie" to describe what you did. But if you want to talk about the movie, you say "I saw a movie last night. It was interesting because.."

Ok I'll add yet another answer, similar to John V Hall's: "watched" implies TV (I would never say I "watched" a movie if it was in a theater). I would use "saw" for either TV or cinema.

I agree with John, except I would add that you can say 'I saw a great movie on tv last night' meaning you were at home. Other comments below have gone off on a tangent about the difference between the words saw and watched which wasn't the question. The question was about the difference between 'saw a movie' and 'watched a movie'.

For me, if I say that I saw a movie, it means that I went to the theater/cinema, regardless of verb tense being used. And if I say that I watched a movie, it means that I did not go anywhere to see the movie, but that I stayed home and watched it on television. We also use "see a show", "see a concert", "see a performance", etc. in a similar way as "see a movie". This is an exception to the usual use of "watch" as something more intentional, and "see" as something more passive.