asperity vs difficulty

asperity and difficulty both are nouns.

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

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

asperity: something hard to endure
difficulty: 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 difficultness.

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