ponytail vs hairstyle

ponytail and hairstyle both are nouns.

Word	Noun	Adjective	Verb	Adverb
ponytail	Yes	No	No	No
hairstyle	Yes	No	No	No

As nouns, hairstyle is a hypernym of ponytail; that is, hairstyle is a word with a broader meaning than ponytail:

ponytail: a hair style that draws the hair back so that it hangs down in back of the head like a pony's tail
hairstyle: the arrangement of the hair (especially a woman's hair)

Other hypernyms of ponytail include coif, coiffure, hair style, hairdo.

ponytail (noun)	hairstyle (noun)
a hair style that draws the hair back so that it hangs down in back of the head like a pony's tail	the arrangement of the hair (especially a woman's hair)