public service vs employment

public service and employment both are nouns.

Word	Noun	Adjective	Verb	Adverb
public service	Yes	No	No	No
employment	Yes	No	No	No

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

public service: employment within a government system (especially in the civil service)
employment: the occupation for which you are paid

Other hypernyms of public service include work.

public service (noun)	employment (noun)
employment within a government system (especially in the civil service)	the occupation for which you are paid
a service that is performed for the benefit of the public or its institutions	the act of using
	the act of giving someone a job
	the state of being employed or having a job