hard time vs difficulty

hard time and difficulty both are nouns.

Word	Noun	Adjective	Verb	Adverb
hard time	Yes	No	No	No
difficulty	Yes	No	No	No

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

hard time: a difficulty that can be overcome with effort
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 hard time include difficultness.

hard time (noun)	difficulty (noun)
a difficulty that can be overcome with effort	an effort that is inconvenient
a term served in a maximum security prison	the quality of being almost beyond one's ability to deal with and requiring a great effort to achieve a positive result