public toilet vs lavatory

public toilet and lavatory both are nouns.

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

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

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

public toilet (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