When would I not want to replace a PN diode with a Schottky diode

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.

If the design has to function over an extended temperature range, leakage becomes a very big issue.