'A few of people' is incorrect. It should be 'few people' or 'a few people' (NB the first of these is negative, and the second positive). 'A handful of' means the same as a few/few, but is more idiomatic and informal. You might say it in conversation or in informal writing, but you wouldn't use it, for example, in an academic report.

There is nit much difference. We don't say "a few of people." We say "few people." This means not many people. "A handful of people" means "a small number of people." Did you bother to look up the words for yourself? That's the best way to learn.

There's almost no difference between "a few people" and "a handful of people." "Few" is straightforward, plain, factual language. "Handful" could be a different way to say the same thing, to add just a little variety or color. To me, "handful" carries the idea of five fingers, so perhaps it suggests a slightly larger number than the word "few." There is an idiomatic expression for a small number, "you could count [something] on the fingers of a hand," and that does, literally, mean five or less. There are absolutely no rules here, but if someone said "I only know the names of a couple of constellations" I would think "two," if he said "I only know the names of a few constellations" I might think "three or four," and if he said "a handful" I might think, oh, five or six.