I’ve wondered the same thing!
On one hand, see and watch can be similar to hear and listen. One is passive (see and hear). While the other is active (watch and listen).
We were watching for whales. But we didn’t see any.
He heard his mom talking, but he really wasn’t listening to what she was saying.
Yet there is another aspect that makes see and watch trickier than hear and listen. And that is the aspect of time.
You watch something when you observe it over a period of time. And you see something when said observation occurs within a window of time.
I think this gets particularly confusing in this day and age since we don’t watch television the same way that we used to watch television. It’s no longer an ongoing spectacle that we tune into. Nowadays, we select a specific program whenever we want to see it.