The difference between Add up and Sum

When used as verbs, add up means to take a sum, whereas sum means to add together.

Sum is also noun with the meaning: a quantity obtained by addition or aggregation.

check bellow for the other definitions of Add up and Sum

Add up as a verb (transitive):

To take a sum.

Examples:

"usex Add up the prices and find out how much it will cost."
Add up as a verb (intransitive):

To accumulate; to amount to.

Examples:

"usex If you can save even a couple of dollars per day, it will add up to a lot over a year."
Add up as a verb (idiomatic, intransitive):

To make sense; to be reasonable or consistent.

Examples:

"usex His story just doesn't add up. Why would he have been at the restaurant the day before the event?"

Sum as a noun:

A quantity obtained by addition or aggregation.

Examples:

"The sum of 3 and 4 is 7."
Sum as a noun (often plural):

An arithmetic computation, especially one posed to a student as an exercise (not necessarily limited to addition).

Examples:

"We're learning about division, and the sums are tricky."
Sum as a noun:

A quantity of money.

Examples:

"a tidy sum"

"the sum of forty pounds"
Sum as a noun:

A summary; the principal points or thoughts when viewed together; the amount; the substance; compendium.

Examples:

"This is the sum of all the evidence in the case."

"This is the sum and substance of his objections."
Sum as a noun:

A central idea or point.
Sum as a noun:

The utmost degree.
Sum as a noun (obsolete):

An old English measure of corn equal to the quarter.