ill health vs sickness

ill health and sickness both are nouns.

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

As nouns, sickness is a hyponym of ill health; that is, sickness 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
sickness: impairment of normal physiological function affecting part or all of an organism

ill health (noun)	sickness (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
	the state that precedes vomiting
	defectiveness or unsoundness