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