sum up vs count

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

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

As verbs, count is a hypernym of sum up; that is, count is a word with a broader meaning than sum up:

Other hypernyms of sum up include enumerate, number, numerate.

sum up (noun)	count (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

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