back down vs pull out

back down and pull out both are verbs.

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

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

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

back down (verb)	pull out (verb)
move backwards from a certain position	remove, usually with some force or effort; also used in an abstract sense
remove oneself from an obligation	move out or away
	bring, take, or pull out of a container or from under a cover
	remove oneself from an obligation