With the verb watch, we are much more active. Watch is like look, but requires more effort from us. We watch things that are going to move, or change in some way. And we watch the movements and changes.
The police decided to watch the suspected murderer rather than arrest him immediately. They hoped he would lead them to the body.
I like watching motor racing on TV.
If you watch that egg for long enough you’ll see it hatch.
We use see to mean simply that an image comes into our eyes. It may not be deliberate. As soon as we open our eyes, we see things.
I can see a cloud in the sky.
I suddenly saw a bird fly in front of me.
Didn’t you see Ram? He was waving at you.
When we look, we try to see. We make a special effort. We concentrate our eyes on something.