Someone or something that is quiet makes only a small amount of noise.
Tania kept the children reasonably quiet and contented...
A quiet murmur passed through the classroom...
The airlines have invested enormous sums in new, quieter aircraft. ≠ noisy
You use quite to indicate that something is the case to a fairly great extent.
Quite is weaker than `very' and `extremely'. = довольно (таки)
I felt quite bitter about it at the time...
Well, actually it requires quite a bit of work and research...
I was quite a long way away, on the terrace...