old woman vs granny

old woman and granny both are nouns.

Word	Noun	Adjective	Verb	Adverb
old woman	Yes	No	No	No
granny	Yes	No	No	No

As nouns, granny is a hyponym of old woman; that is, granny is a word with a more specific, narrower meaning than old woman:

Other hyponyms of old woman include beldam, beldame, beldam, beldame, crone, hag, witch, mother.

old woman (noun)	granny (noun)
a woman who is old	a reef knot crossed the wrong way and therefore insecure
herb with greyish leaves found along the east coast of North America; used as an ornamental plant	the mother of your father or mother
	an old woman