public convenience vs lavatory

public convenience and lavatory both are nouns.

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

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

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

public convenience (noun)	lavatory (noun)
a toilet that is available to the public	a room or building equipped with one or more toilets
	a bathroom sink that is permanently installed and connected to a water supply and drainpipe; where you can wash your hands and face
	a toilet that is cleaned of waste by the flow of water through it