count vs add together

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

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

As verbs, add together is a hyponym of count; that is, add together is a word with a more specific, narrower meaning than count:

Other hyponyms of count include recount, miscount, census, add, add up, sum, sum up, summate, tally, tot, tot up, total, tote up.

count (noun)	add together (noun)
the act of counting; reciting numbers in ascending order
a nobleman (in various countries) having rank equal to a British earl
the total number counted

count (verb)	add together (verb)
include as if by counting	make an addition by combining numbers
have faith or confidence in	determine the sum of
take account of
determine the number or amount of
name or recite the numbers in ascending order
show consideration for; take into account
have weight; have import, carry weight
put into a group
have a certain value or carry a certain weight