add up vs sum

add up is not a noun while sum is a noun.

Word	Noun	Adjective	Verb	Adverb
add up	No	No	Yes	No
sum	Yes	No	Yes	No

As verbs, add up and sum are synonyms defined as:

Other synonyms of add up include add, add together, sum up, summate, tally, tot, tot up, total, tote up.

add up (noun)	sum (noun)
	the whole amount
	the choicest or most essential or most vital part of some idea or experience
	a quantity obtained by the addition of a group of numbers
	the final aggregate
	a set containing all and only the members of two or more given sets
	a quantity of money

add up (verb)	sum (verb)
determine the sum of	determine the sum of
be reasonable or logical or comprehensible	be a summary of
to measure up to in kind or quality
add up in number or quantity