public convenience vs facility

public convenience and facility both are nouns.

Word	Noun	Adjective	Verb	Adverb
public convenience	Yes	No	No	No
facility	Yes	No	No	No

As nouns, facility is a hypernym of public convenience; that is, facility is a word with a broader meaning than public convenience:

public convenience: a toilet that is available to the public
facility: something designed and created to serve a particular function and to afford a particular convenience or service

Other hypernyms of public convenience include bathroom, can, john, lav, lavatory, privy, toilet.

public convenience (noun)	facility (noun)
a toilet that is available to the public	a service that an organization or a piece of equipment offers you
	something designed and created to serve a particular function and to afford a particular convenience or service
	a building or place that provides a particular service or is used for a particular industry
	a natural effortlessness
	skillful performance or ability without difficulty