woman of the house vs lady of the house

woman of the house and lady of the house both are nouns.

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

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

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

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

woman of the house (noun)	lady 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