mental hygiene vs psychiatry

mental hygiene and psychiatry both are nouns.

Word	Noun	Adjective	Verb	Adverb
mental hygiene	Yes	No	No	No
psychiatry	Yes	No	No	No

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

mental hygiene: the branch of psychiatry concerned with psychological methods
psychiatry: the branch of medicine dealing with the diagnosis and treatment of mental disorders

Other hypernyms of mental hygiene include psychological medicine, psychopathology.

mental hygiene (noun)	psychiatry (noun)
the branch of psychiatry concerned with psychological methods	the branch of medicine dealing with the diagnosis and treatment of mental disorders