longhand vs handwriting

longhand and handwriting both are nouns.

Word	Noun	Adjective	Verb	Adverb
longhand	Yes	Yes	No	No
handwriting	Yes	No	No	No

As nouns, handwriting is a hypernym of longhand; that is, handwriting is a word with a broader meaning than longhand:

longhand: rapid handwriting in which letters are set down in full and are cursively connected within words without lifting the writing implement from the paper
handwriting: something written by hand

Other hypernyms of longhand include hand, script.

longhand (noun)	handwriting (noun)
rapid handwriting in which letters are set down in full and are cursively connected within words without lifting the writing implement from the paper	the activity of writing by hand
	something written by hand

longhand (adjective)	handwriting (adjective)
having words written out in full by hand