house husband vs hubby

house husband and hubby both are nouns.

Word	Noun	Adjective	Verb	Adverb
house husband	Yes	No	No	No
hubby	Yes	No	No	No

As nouns, hubby is a hypernym of house husband; that is, hubby is a word with a broader meaning than house husband:

house husband: a husband who keeps house while his wife earns the family income
hubby: a married man; a woman's partner in marriage

Other hypernyms of house husband include husband, married man.

house husband (noun)	hubby (noun)
a husband who keeps house while his wife earns the family income	a married man; a woman's partner in marriage