hair care vs hairdressing

hair care and hairdressing both are nouns.

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

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

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

Other synonyms of hair care include haircare.

hair care (noun)	hairdressing (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