playwright vs author

playwright and author both are nouns.

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

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

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

Other hypernyms of playwright include writer.

playwright (noun)	author (noun)
someone who writes plays	writes (books or stories or articles or the like) professionally (for pay)
	someone who originates or causes or initiates something

playwright (verb)	author (verb)
	be the author of