sum vs add together

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

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

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

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

sum (noun)	add together (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

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