The main difference is that in the first example, time is being used as a non-count noun. It's being treated as infinitely divisible. Time has a count noun usage where it's a synonym for event or occurrence but that's not the meaning of time being used here.
On the other hand, moment is a count noun. We can have one moment, a few moments, or various other usages that treat a moment as a countable object.
Because of this difference, we talk about "a moment" that we either have or don't have; but we talk about "some time" or "no time" if we haven't got any.
Duolingo claims that one can not use "NO" in the second sentence instead of "NOT" and I would like to know why.
Duolingo is correct.
We can use "no" as a kind of indefinite article (like German kein) indicating lack of something. We can't use this "indefinte article" in combination with another definite article.