Skillful is an adjective which modifies a noun or pronoun; skillfulness is a noun which can be used as a subject in a sentence, a direct object, a predicate nominative, etc.
He is very skillful at drawing with charcoal. [adjective, predicate adjective]
His skillfulness in using charcoal as an instrument of art is apparent to me. [noun, subject]
He skillfully and casually moves across the canvass to darken those areas needing the most charcoal. [adverb, modifies the verb "moves"]
So, what you do is look up the part of speech used as the main entry, and then with the other definitions for the same word, study what part of speech they are. This will determine how the words are used in sentences.
For another example:
She is angry about her grades. [adjective, predicate adjective]
She feels her angriness is justified by the fact she studied so hard. [noun, "...(that) her angriness is..." subject of a dependent clause]
A review of the eight parts of speech:
"He demonstrated great skillfulness" seems merely a clunkier way of
saying "He demonstrated great skill."
Both are nouns. So what’s the difference?
You just said it.
This happens all the time with words; the right words in the not so right way; a popular definition is: awkward; another is redundant.
You would use “skillfulness” in one sentence to express an idea clearly, and “skill” in another to express a different idea.
He demonstrated great skill. I wonder how he got to be so skillful at what he does. It’s obvious that his skillfulness took many years to develop.