I'm going to use a sunrise to explain the difference. Look at the following sentences:
1. I saw the sunrise this morning.
2. I watched the sunrise this morning.
In sentence 1, 'saw' means that you looked up at the sky while the sun was rising. For an instant, you looked at the sunrise, but then you continued what you were doing.
In sentence 2, 'watched' means that you looked at the sunrise for the duration of the event. Sunrises usually take around 3-5 minutes to 'watch', but only a second to 'see'.
There are two major expections with 'see' that I can think of:
1. I saw a movie last night.
This is a very common usage of 'see' that does not agree with how we usually use it.
2. I saw my mom last week.
In cases like this, 'see' is used to mean 'visit', so it doesn't follow the pattern, either.
I hope this helps. I know it's a confusing thing to understand.