spelling vs writing system

spelling and writing system both are nouns.

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

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

spelling: forming words with letters according to the principles underlying accepted usage
writing system: a method of representing the sounds of a language by written or printed symbols

Other hypernyms of spelling include orthography.

spelling (noun)	writing system (noun)
forming words with letters according to the principles underlying accepted usage	a method of representing the sounds of a language by written or printed symbols