Regular alpha is based on covariances.
Standardized alpha is based on correlations (which can be thought of as based on standardized covariances, or standardized variables).
Standardizing the variables makes the variances equal to 1.00 - you don't actually need to standardize the variables, as long as you make the variances all equal (to any value) you will get standardized alpha.
Usually we create a scale score by summing the scores on the items. In this case, alpha is OK.
But, if you have measures with very different scales - e.g. one test is on a 1-7 scale, one is on a 1-100 scale, it doesn't make sense to add these scores together - you should do something to put them on the same scale - like standardizing the variables first. In this case you should use standardized alpha.
The second case is unusual, and so you usually use alpha (not standardized alpha). In addition, items usually have similar variances to start with, so the differences between alpha and standardized alpha are (usually) small.