See means to use your visual sense. "There are those who have eyes, but see not." Watch implies seeing for an extended period, to learn something.
Reminds me of Army training for enemy aircraft recognition. Pictures were shown repeatedly, for increasingly short intervals. Finally the Intervals got down to 1/25th of a second and less. The guys loudly protesteed that nothing could be identified in such a short interval. The projectionist then showed a nude pinup at 1/60th second. Nobody had trouble recongizine what that was.