illness vs ill health

illness and ill health both are nouns.

Word	Noun	Adjective	Verb	Adverb
illness	Yes	No	No	No
ill health	Yes	No	No	No

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

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

Other hypernyms of illness include health problem, unhealthiness.

illness (noun)	ill health (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