public housing vs housing project

public housing and housing project both are nouns.

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

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

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

public housing (noun)	housing project (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