add together vs sum

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

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

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

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

add together (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 together (verb)	sum (verb)
make an addition by combining numbers	determine the sum of
determine the sum of	be a summary of