patriarch vs head of household

patriarch and head of household both are nouns.

Word	Noun	Adjective	Verb	Adverb
patriarch	Yes	No	No	No
head of household	Yes	No	No	No

As nouns, head of household is a hypernym of patriarch; that is, head of household is a word with a broader meaning than patriarch:

Other hypernyms of patriarch include adult male, man.

patriarch (noun)	head of household (noun)
the male head of family or tribe	the head of a household or family or tribe
any of the early biblical characters regarded as fathers of the human race
title for the heads of the Eastern Orthodox Churches (in Istanbul and Alexandria and Moscow and Jerusalem)
a man who is older and higher in rank than yourself