‘See’ and ‘watch’ are words that are commonly misunderstood as being interchangeable, but while both refer to the optic function, they are not one and the same thing. They stand for different ways of using the eye.
‘To see’ something is simply to look at it or spot it; it may be unintentional, ie, you may not have been looking for it. Example : I saw a man up ahead.
‘To watch’ something is look at something closely or intently, generally because it is moving. Example : I watched the movie.