In standarization, technically, there is no limit as to what value can the extremes of output have. In case of normalization, however, the most extreme values will be equal to 1 and -1.
This will suppress the effect of outliers. Why?
For example lets consider a situation when in dataset feature A has a very large variance and feature B has very small (i.e. we expect to find a value of A far from mean quite often, but finding a value of B far from mean is something very unusual). This is a vital information, as finding an example with a very extreme value of B might suggest that this example is an anomaly/ positive/ negative or whatever you want to detect, and finding an extreme value of A is not a very outstanding thing.
After standarization, the extreme values of B will be much further from the zero than values of A, whereas after normalization, in both cases the most extreme values will be equal to 1 and -1. Therefore, the information that an extreme value of B is much more important than an extreme value of A is lost, because in the end, values of these extremes will be equal.