hairdressing vs hair care

hairdressing and hair care both are nouns.

Word	Noun	Adjective	Verb	Adverb
hairdressing	Yes	No	No	No
hair care	Yes	No	No	No

As nouns, hairdressing and hair care are synonyms defined as:

hairdressing and hair care: care for the hair: the activity of washing or cutting or curling or arranging the hair

Other synonyms of hairdressing include haircare.

hairdressing (noun)	hair care (noun)
care for the hair: the activity of washing or cutting or curling or arranging the hair	care for the hair: the activity of washing or cutting or curling or arranging the hair
a toiletry for the hair