hair style vs hairdo

hair style and hairdo both are nouns.

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

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

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

Other synonyms of hair style include coif, coiffure, hairstyle.

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