housing project vs public housing

housing project and public housing both are nouns.

Word	Noun	Adjective	Verb	Adverb
housing project	Yes	No	No	No
public housing	Yes	No	No	No

As nouns, housing project and public housing are synonyms defined as:

housing project and public housing: a housing development that is publicly funded and administered for low-income families

housing project (noun)	public housing (noun)
a housing development that is publicly funded and administered for low-income families	a housing development that is publicly funded and administered for low-income families