script vs writing system

script and writing system both are nouns.

Word	Noun	Adjective	Verb	Adverb
script	Yes	No	Yes	No
writing system	Yes	No	No	No

As nouns, writing system is a hypernym of script; that is, writing system is a word with a broader meaning than script:

script: a particular orthography or writing system
writing system: a method of representing the sounds of a language by written or printed symbols

Other hypernyms of script include orthography.

script (noun)	writing system (noun)
a written version of a play or other dramatic composition; used in preparing for a performance	a method of representing the sounds of a language by written or printed symbols
a particular orthography or writing system
something written by hand

script (verb)	writing system (verb)
write a script for