Electronic – Schottky Diodes Depletion Region

First it depends on the type of semiconductor(SC) n and p types are different. I'll just do n-type. (p-type is the mirror image of n-type). Then, in theory, it only depends on the relative work function of the metal and the SC. If the metal is below the SC then electrons can flow and no diode. But if the metal is above the SC then you get a depletion region in the SC, and a diode.

Oh, in practice other things, like surface states play a role. (At least that's my limited understanding.)

While Schottky diodes generally have a lower forward voltage drop and faster recovery time, they are also more susceptible to reverse surges. In my experience a Schottky diode will fail at a lower reverse voltage (or during a transient) that a standard silicon diode would handle without a problem. It all comes down to your application, what the diode is doing, and what sort of electrical events you expect it to see. There is no right or wrong answer here.