public service vs community service

public service and community service both are nouns.

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

As nouns, public service and community service are synonyms defined as:

public service and community service: a service that is performed for the benefit of the public or its institutions

public service (noun)	community service (noun)
employment within a government system (especially in the civil service)	an unpaid service for the benefit of the public that is performed by lawbreakers as part (or all) of their sentence
a service that is performed for the benefit of the public or its institutions	a service that is performed for the benefit of the public or its institutions