half brother vs half-brother

half brother and half-brother both are nouns.

Word	Noun	Adjective	Verb	Adverb
half brother	Yes	No	No	No
half-brother	Yes	No	No	No

As nouns, half brother and half-brother are synonyms defined as:

half brother and half-brother: a brother who has only one parent in common with you

Other synonyms of half brother include stepbrother.

half brother (noun)	half-brother (noun)
a brother who has only one parent in common with you	a brother who has only one parent in common with you