It depends on the context. We use both verbs, the difference being that 'watch' refers specifically to the activity of sitting in front of a screen and focusing on the film being shown. 'See' is rather more general. Here are some examples:
"Hi! Can I call you back later? I'm just watching a really good film on TV and it's nearly finished. I'll call you back in about ten minutes, OK?"
Here we have to use 'watch', because you're referring to the activity that you're engaged in at the moment.
"What's the best movie that you've ever seen?"
Here we use 'see' because the focus is on your familiarity with the film rather than on the activity of looking at the screen.
In some cases, you can use either verb:
"Do you want to see a movie tonight? ( or should we go ten-pin bowling instead?) "
This suggests an outing, involving a trip to the cinema. The focus is more on the whole evening's activity.
"Do you want to watch a movie tonight? ( or should we watch our favourite TV miniseries? )"
This suggests that the speakers are at home, with more of an emphasis on the activity of watching.
I think that our usage of 'see' comes from the days before digital entertainment - before Netflix, before the internet, before TV on demand, before even video recorders. For much of the 20th century, you could only see films at the moment when they were being shown at the cinema, and later if they were being shown on TV. They were there, and then they were gone. If you didn't see a particular film at the time when it was 'on', it was too late. The use of the verb 'see' emphasises the idea of a fleeting chance to catch something before it disappeared.