despair vs hopelessness

despair and hopelessness both are nouns.

Word	Noun	Adjective	Verb	Adverb
despair	Yes	No	Yes	No
hopelessness	Yes	No	No	No

As nouns, hopelessness is a hyponym of despair; that is, hopelessness is a word with a more specific, narrower meaning than despair:

despair: the feeling that everything is wrong and nothing will turn out well
hopelessness: the despair you feel when you have abandoned hope of comfort or success

Other hyponyms of despair include resignation, surrender, discouragement, disheartenment, dismay, pessimism.

despair (noun)	hopelessness (noun)
the feeling that everything is wrong and nothing will turn out well	the despair you feel when you have abandoned hope of comfort or success
a state in which all hope is lost or absent

despair (verb)	hopelessness (verb)
abandon hope; give up hope; lose heart