writing vs handwriting

writing and handwriting both are nouns.

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

As nouns, handwriting is a hyponym of writing; that is, handwriting is a word with a more specific, narrower meaning than writing:

Other hyponyms of writing include coding, cryptography, secret writing, steganography, inscription, notation.

As nouns, handwriting is a hyponym of writing; that is, handwriting is a word with a more specific, narrower meaning than writing:

writing: letters or symbols that are written or imprinted on a surface to represent the sounds or words of a language
handwriting: something written by hand

writing (noun)	handwriting (noun)
the activity of putting something in written form	the activity of writing by hand
the act of creating written works	something written by hand
the work of a writer; anything expressed in letters of the alphabet (especially when considered from the point of view of style and effect)
letters or symbols that are written or imprinted on a surface to represent the sounds or words of a language
(usually plural) the collected work of an author