rector vs man of the cloth

rector and man of the cloth both are nouns.

Word	Noun	Adjective	Verb	Adverb
rector	Yes	No	No	No
man of the cloth	Yes	No	No	No

As nouns, man of the cloth is a hypernym of rector; that is, man of the cloth is a word with a broader meaning than rector:

rector: a person authorized to conduct religious worship
man of the cloth: a member of the clergy and a spiritual leader of the Christian Church

Other hypernyms of rector include clergyman, reverend.

rector (noun)	man of the cloth (noun)
a person authorized to conduct religious worship	a member of the clergy and a spiritual leader of the Christian Church