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