writing style vs prose

writing style and prose both are nouns.

Word	Noun	Adjective	Verb	Adverb
writing style	Yes	No	No	No
prose	Yes	No	No	No

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

Other hyponyms of writing style include drama, form, poesy, poetry, verse, satire, steampunk.

writing style (noun)	prose (noun)
a style of expressing yourself in writing	ordinary writing as distinguished from verse
	matter of fact, commonplace, or dull expression