mental health vs sanity

mental health and sanity both are nouns.

Word	Noun	Adjective	Verb	Adverb
mental health	Yes	No	No	No
sanity	Yes	No	No	No

As nouns, sanity is a hyponym of mental health; that is, sanity is a word with a more specific, narrower meaning than mental health:

mental health: the psychological state of someone who is functioning at a satisfactory level of emotional and behavioral adjustment
sanity: normal or sound powers of mind

Other hyponyms of mental health include mental balance, mental soundness, saneness.

mental health (noun)	sanity (noun)
the psychological state of someone who is functioning at a satisfactory level of emotional and behavioral adjustment	normal or sound powers of mind