What is the difference between normalization and re-scaling?

Re-scaling means that you multiply your variable by a factor, i.e. $x to x/a$. A normalization is a specific kind of re-scaling, where the factor $a$ is such that the values of $x$ become of order one. Its form depends on the context and what you are trying to do. Examples are $a= langle x rangle$ or $a= textrm{max}(x)$. Perhaps the most common one is when you do $x to (x-langle x rangle)/sigma$, because if $x$ follows a Gaussian distribution, the re-scaled variable follows a normal distribution (average 0 and std 1). Note, this last transformation actually involves a re-scaling, but also a translation.