$Ax = b$ & $Ax + b$

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Ask a dumb question but confuse me long time.

The following is what I know:

1st case
$Ax = b$ is an affine set in $x$,i.e. $\{x | Ax = b\}$, and it is linear in $x$.
2nd case
$ f(x) = Ax + b$ is an affine function. It is not linear because of $b$.But $f(x) = Ax$ is linear in $x$.

My question is I am still confused about both of them, can anyone provide a more mature way to seperate both of them? What is the main difference (or application) between them?