back up vs boost

back up is not a noun while boost is a noun.

Word	Noun	Adjective	Verb	Adverb
back up	No	No	Yes	No
boost	Yes	No	Yes	No

As verbs, boost is a hyponym of back up; that is, boost is a word with a more specific, narrower meaning than back up:

back up (noun)	boost (noun)
	the act of giving hope or support to someone
	the act of giving a push
	an increase in cost

back up (verb)	boost (verb)
establish as valid or genuine	increase or raise
become or cause to become obstructed	give a boost to; be beneficial to
make a copy of (a computer file) especially for storage in another place as a security copy	push or shove upward, as if from below or behind
move backwards from a certain position	increase
give moral or psychological support, aid, or courage to	contribute to the progress or growth of