sickness vs health problem

sickness and health problem both are nouns.

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

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

sickness: 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 sickness include ill health, unhealthiness.

sickness (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
the state that precedes vomiting
defectiveness or unsoundness