asperity vs difficultness

asperity and difficultness both are nouns.

Word	Noun	Adjective	Verb	Adverb
asperity	Yes	No	No	No
difficultness	Yes	No	No	No

As nouns, difficultness is a hypernym of asperity; that is, difficultness is a word with a broader meaning than asperity:

asperity: something hard to endure
difficultness: the quality of being almost beyond one's ability to deal with and requiring a great effort to achieve a positive result

Other hypernyms of asperity include difficulty.

asperity (noun)	difficultness (noun)
something hard to endure	the quality of being almost beyond one's ability to deal with and requiring a great effort to achieve a positive result
harshness of manner