handwriting vs script

handwriting and script both are nouns.

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

As nouns, handwriting and script are synonyms defined as:

Other synonyms of handwriting include hand.

handwriting (noun)	script (noun)
the activity of writing by hand	a written version of a play or other dramatic composition; used in preparing for a performance
something written by hand	a particular orthography or writing system
	something written by hand

handwriting (verb)	script (verb)
	write a script for