playwright vs writer

playwright and writer both are nouns.

Word	Noun	Adjective	Verb	Adverb
playwright	Yes	No	No	No
writer	Yes	No	No	No

As nouns, writer is a hypernym of playwright; that is, writer is a word with a broader meaning than playwright:

playwright: someone who writes plays
writer: writes (books or stories or articles or the like) professionally (for pay)

Other hypernyms of playwright include author.

playwright (noun)	writer (noun)
someone who writes plays	writes (books or stories or articles or the like) professionally (for pay)
	a person who is able to write and has written something