illness vs health problem

illness and health problem both are nouns.

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

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

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

Other hypernyms of illness include ill health, unhealthiness.

illness (noun)	health problem (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