count vs add up

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

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

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

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

count (noun)	add 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)	add up (verb)
include as if by counting	determine the sum of
have faith or confidence in	be reasonable or logical or comprehensible
take account of	to measure up to in kind or quality
determine the number or amount of	add up in number or quantity
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