hairstyle vs hair style

hairstyle and hair style both are nouns.

Word	Noun	Adjective	Verb	Adverb
hairstyle	Yes	No	No	No
hair style	Yes	No	No	No

As nouns, hairstyle and hair style are synonyms defined as:

hairstyle and hair style: the arrangement of the hair (especially a woman's hair)

Other synonyms of hairstyle include coif, coiffure, hairdo.

hairstyle (noun)	hair style (noun)
the arrangement of the hair (especially a woman's hair)	the arrangement of the hair (especially a woman's hair)