mental health vs mental state

mental health and mental state both are nouns.

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

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

mental health: the psychological state of someone who is functioning at a satisfactory level of emotional and behavioral adjustment
mental state: (psychology) a mental condition in which the qualities of a state are relatively constant even though the state itself may be dynamic

Other hypernyms of mental health include mental condition, psychological condition, psychological state.

mental health (noun)	mental state (noun)
the psychological state of someone who is functioning at a satisfactory level of emotional and behavioral adjustment	(psychology) a mental condition in which the qualities of a state are relatively constant even though the state itself may be dynamic