count vs sum up

count is a noun but sum up is not a noun.

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

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

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

count (noun)	sum up (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)	sum up (verb)
include as if by counting	give a summary (of)
have faith or confidence in	determine the sum of
take account of	be a summary 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