ill health vs illness

ill health and illness both are nouns.

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

As nouns, illness is a hyponym of ill health; that is, illness is a word with a more specific, narrower meaning than ill health:

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

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