These verbs both refer to visual activity, but can’t be used interchangeably.
To ‘see’ is a verb of perception. To see is an effortless action. You don’t necessarily plan or intend to ‘see’ something, it just happens.
‘I saw a bird nest in the tree next to our house.’
In this example, the speaker has come across the nest by chance; they did not go out of their way to see it.
To ‘watch’ is to look at something on purpose. It is a voluntary action; you are intently focused on watching an event.
‘Please close the curtains, I’m watching the football.’
In this example, the speaker is currently ‘watching’ something that is in motion, and they are doing so on purpose.
Source : Learn English