add together vs sum up

add together and sum up both are verbs.

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

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

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

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