In general I'd agree with TheStephen. The only thing I'd add is
that "watch" is rather more active than "see". When you watch
something you make a conscious effort to see it. (Like the
difference between "listen" and "hear".) In general, we see a
performance in public and watch a performance on TV. So for
example: - We're going to see a movie at the cinema tonight. -
We're going to watch a movie on TV tonight. Here is a page about
Watch versus See versus Look