English is my language. I have to say that that I don't know the difference. To me they are just different ways of saying the same thing. Both present tense. There may be an expert out there who can give a a technical difference but for my everyday spoken UK English, there is no difference.

The grammar is a bit different, but it has the exact same meaning. Consider it a style choice and neither is better

The above comments are the best answers here!

BTW: In general "the child" sounds rather formal/impersonal. I have difficulty seeing a case when this expression would feel the most natural to use. I'll watch the cat while you sleep. I'll watch young Jimmy while you sleep. I'll watch the baby while you sleep. <--- seems ok I'll watch the child while you sleep. <-- feels funny

Maybe in general "you <verb>" & "you are <verb>ing" differ in terms of the second one stressing the continued act, the duration of doing something. Here, with "while" you are naturally talking about the period while something is done, so they come out to be the same thing.