In many cases they're interchangeable, but if you say you watched someone "do" something, it often means you watched the whole process from beginning to end (you watched the person do the action and complete it). If you watched them "doing" something, you watched them *while* they were doing the thing, but you didn't necessarily watch them the whole time. "I watched him walk up the stairs." -- The sentence implies that he walked all the way to the top of the stairs, and I watched him the whole time. "I watched him walking up the stairs." -- I was watching for at least part of the time while he was walking up the stairs.