ill health vs health problem

ill health and health problem both are nouns.

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

As nouns, ill health and health problem are synonyms defined as:

ill health and health problem: a state in which you are unable to function normally and without pain

Other synonyms of ill health include unhealthiness.

ill health (noun)	health problem (noun)
a state in which you are unable to function normally and without pain	a state in which you are unable to function normally and without pain