back down vs bow out

back down and bow out both are verbs.

Word	Noun	Adjective	Verb	Adverb
back down	No	No	Yes	No
bow out	No	No	Yes	No

As verbs, back down and bow out are synonyms defined as:

Other synonyms of back down include back off, chicken out, pull out.

back down (verb)	bow out (verb)
move backwards from a certain position	retire gracefully
remove oneself from an obligation	remove oneself from an obligation