sickness vs ill health

sickness and ill health both are nouns.

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

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

sickness: 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 sickness include health problem, unhealthiness.

sickness (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
the state that precedes vomiting
defectiveness or unsoundness