In short, there is not much difference. In English, adjectives and nouns and verbs are not distinguished by their spelling, but only by their position in a sentence. So a word "quiet" might be an adjective, verb or noun.
Now adjectives can often form nouns by adding "-ness", so we have "noisy" and "noisiness". As there is no noun "noisy" the word "noisiness" is not redundant. The noun "noise" has a different meaning from "noisiness"
We can form the noun "quietness" in the same way, but in doing so it happens to have almost the same meaning as the noun quiet.
There is a slight distinction:
The quiet of the wood
Might mean the quiet part of the wood. Other parts of the wood might not be quiet. It follows the pattern "The quiet of the night", compare with "The centre of the wood".
The quietness of the wood
Means that the wood is quiet, and I'm discussing this property of the wood.
Walking further from the town, John entered the quiet of the wood. Here the only sounds were the calls of the birds and the rustle of the leaves.
The quietness of the wood was broken by a helicopter flying low over the trees.