illness vs unhealthiness

illness and unhealthiness both are nouns.

Word	Noun	Adjective	Verb	Adverb
illness	Yes	No	No	No
unhealthiness	Yes	No	No	No

As nouns, unhealthiness is a hypernym of illness; that is, unhealthiness is a word with a broader meaning than illness:

illness: impairment of normal physiological function affecting part or all of an organism
unhealthiness: a state in which you are unable to function normally and without pain

Other hypernyms of illness include health problem, ill health.

illness (noun)	unhealthiness (noun)
impairment of normal physiological function affecting part or all of an organism	a state in which you are unable to function normally and without pain