public square vs market place

public square and market place both are nouns.

Word	Noun	Adjective	Verb	Adverb
public square	Yes	No	No	No
market place	Yes	No	No	No

As nouns, market place is a hypernym of public square; that is, market place is a word with a broader meaning than public square:

public square: a place of assembly for the people in ancient Greece
market place: an area in a town where a public mercantile establishment is set up

Other hypernyms of public square include market, marketplace, mart.

public square (noun)	market place (noun)
a place of assembly for the people in ancient Greece	the world of commercial activity where goods and services are bought and sold
an open area at the meeting of two or more streets	an area in a town where a public mercantile establishment is set up