rude vs bad-mannered

rude and bad-mannered both are adjectives.

Word	Noun	Adjective	Verb	Adverb
rude	No	Yes	No	No
bad-mannered	No	Yes	No	No

As adjectives, rude and bad-mannered are synonyms defined as:

Other synonyms of rude include ill-mannered, unmannered, unmannerly.

rude (adjective)	bad-mannered (adjective)
lacking civility or good manners	socially incorrect in behavior
belonging to an early stage of technical development; characterized by simplicity and (often) crudeness
socially incorrect in behavior
(used especially of commodities) being unprocessed or manufactured using only simple or minimal processes
(of persons) lacking in refinement or grace