Correct usage of "see" vs. "watch"

1. I have seen them grow up.
2. I have watched them grow up.

Though the intended meaning is conveyed in both sentences, I want to know which in this case is a better fit, see or watch.

Solution 1:

(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.

Solution 2:

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.")

Solution 3:

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.