pied-a-terre vs housing

pied-a-terre and housing both are nouns.

Word	Noun	Adjective	Verb	Adverb
pied-a-terre	Yes	No	No	No
housing	Yes	No	No	No

As nouns, housing is a hypernym of pied-a-terre; that is, housing is a word with a broader meaning than pied-a-terre:

Other hypernyms of pied-a-terre include living accommodations, lodging.

pied-a-terre (noun)	housing (noun)
lodging for occasional or secondary use	structures collectively in which people are housed
	a protective cover designed to contain or support a mechanical component
	stable gear consisting of a decorated covering for a horse, especially (formerly) for a warhorse