Correct usage of "see" vs. "watch"

(text lifted from John Lawler's comment - he probably won't post it himself, but it's solid stuff)

The basic semantic difference between see and watch (and for that matter, look) is that see is non-volitional, while look and watch are volitional; additionally, watch is durative, i.e, it means that the looking went on for a while. In this context, the best fit is a matter of what you want to imply, and what you want to state, and that's up to the speaker in context.

Watch has the connotation of both duration ("watch a movie") and vigilance ("watch your back!")

See typically means simply to successfully observe ("I see you!") There is a secondary meaning of "investigate" ("I'm going to see what's up.")

There are supportable arguments for the use of either word. The sentence is meant to convey that the speaker has witnessed a process, "growing up". "Seen" can be used in relation to an event that occurs in an instant or or one that occurs over a period of time. However, "watched" is appropriate only in relation to an event that occurs over a period of time. Therefore, "watched" would be the better choice here.