It has been a while since I watched you sleep.
Is watched or saw more correct in sentence above? What is the difference in meaning between the two choices?
I’ll address ‘watch’ and ‘see’.
In one line – watch is for moving object or looking at/seeing something over a period of time. Generally the result is expected. see is just seeing, in most of the cases, still objects and it might not be intentional.
Watch generally refers to something which is in motion and you are interested in the result. See simply implies looking the still subject. Also see comes with no intention and watch comes with intention. In other words, seeing is not avoidable but watch is intentionally done. You open up the curtain and see the birds outside. You watch birds drinking water (moving, result oriented).
You generally don’t see movie on television but watch it. On the other hand you generally don’t watch still photographs but see/look (at) them.
When we watch something, we intend to look at it but we’re also looking at it quite intensely, usually because it’s moving. So, for example, "I watched the bus go through the traffic lights." "I watched the movie." We want to see it, we’re looking at it intensely and it’s normally moving.
Remember, ‘see’ – you didn’t really intend to, it just happened; ‘look’ – you intended to do it; and watch you intended to do it and you were looking intensely, usually because it was moving.
Now it depends on the context which way sleep is used. Do you mean that you observed/watched the object trying to fall asleep and then finally slept? If that so, I watch might work. But still, to my ears, watching someone sleep does not look natural unless you do that for a period of time. Better expression to say that you are looking at a sleeping person for quite some time (as for a gatekeeper), you may say…
I have been watching you sleeping for past half an hour
On the other hand, saw can be used this way…
I saw you sleeping for half an hour