chemist's shop vs drugstore

chemist's shop and drugstore both are nouns.

Word	Noun	Adjective	Verb	Adverb
chemist's shop	Yes	No	No	No
drugstore	Yes	No	No	No

As nouns, chemist's shop and drugstore are synonyms defined as:

chemist's shop and drugstore: a retail shop where medicine and other articles are sold

Other synonyms of chemist's shop include apothecary's shop, chemist's, pharmacy.

chemist's shop (noun)	drugstore (noun)
a retail shop where medicine and other articles are sold	a retail shop where medicine and other articles are sold