The difference between Absolute value and Norm
When used as nouns, absolute value means for a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, \sqrt{a^2+b^2}. denoted by  , whereas norm means that which is regarded as normal or typical.
Norm is also verb with the meaning: to endow (a vector space, etc) with a norm.
check bellow for the other definitions of Absolute value and Norm

Absolute value as a noun (mathematics):
For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, \sqrt{a^2+b^2}. Denoted by  .
Examples:
"The absolute value x of a real number x is <math>\sqrt{x^2}</math>, which is equal to x if x is nonnegative, and −x if x is negative."

Norm as a noun (usually, definite, '''the norm'''):
That which is regarded as normal or typical.
Examples:
"Unemployment is the norm in this part of the country."

Norm as a noun:
A rule that is enforced by members of a community.
Examples:
"Not eating your children is just one of those societal norms."

Norm as a noun (philosophy, computer science):
A sentence with nondescriptive meaning, such as a command, permission or prohibition.

Norm as a noun (mathematics):
A function, generally denoted v\mapsto\leftv\right or v\mapsto\left\v\right\, that maps vectors to nonnegative scalars and has the following properties: if v\ne0 then \left\v\right\\ne0; given a scalar k, \left\kv\right\=\leftk\right\cdot\left\v\right\, where \leftk\right is the absolute value of k; given two vectors v,w, \left\v+w\right\\le\left\v\right\+\left\w\right\ (the triangle inequality).

Norm as a noun (chess):
A high level of performance in a chess tournament, several of which are required for a player to receive a title.

Norm as a verb (analysis):
To endow (a vector space, etc) with a norm.