carry on vs hang on

carry on and hang on both are verbs.

Word	Noun	Adjective	Verb	Adverb
carry on	No	No	Yes	No
hang on	No	No	Yes	No

As verbs, hang on is a hyponym of carry on; that is, hang on is a word with a more specific, narrower meaning than carry on:

carry on (verb)	hang on (verb)
continue talking	be persistent, refuse to stop
direct the course of; manage or control	hold the phone line open
misbehave badly; act in a silly or improper way	fix to; attach
keep or maintain in unaltered condition; cause to remain or last