housewife vs woman of the house

housewife and woman of the house both are nouns.

Word	Noun	Adjective	Verb	Adverb
housewife	Yes	No	No	No
woman of the house	Yes	No	No	No

As nouns, housewife and woman of the house are synonyms defined as:

housewife and woman of the house: a wife who manages a household while her husband earns the family income

Other synonyms of housewife include homemaker, lady of the house.

housewife (noun)	woman of the house (noun)
a wife who manages a household while her husband earns the family income	a wife who manages a household while her husband earns the family income