hairdo vs hair style

hairdo and hair style both are nouns.

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

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

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

Other synonyms of hairdo include coif, coiffure, hairstyle.

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