half-staff vs half-mast

half-staff and half-mast both are nouns.

Word	Noun	Adjective	Verb	Adverb
half-staff	Yes	No	No	No
half-mast	Yes	No	No	No

As nouns, half-staff and half-mast are synonyms defined as:

half-staff and half-mast: a position some distance below the top of a mast to which a flag is lowered in mourning or to signal distress

half-staff (noun)	half-mast (noun)
a position some distance below the top of a mast to which a flag is lowered in mourning or to signal distress	a position some distance below the top of a mast to which a flag is lowered in mourning or to signal distress