The downside is that Schottky does not have as sharp an IV curve. That is, higher currents make the voltage drop a bit higher. While it turns on at 0.3v, at high currents a Schottky can drop more voltage than a std rectifier, up to 1v or more. This is a somewhat inaccurate generalization however since it depends on the rated current of the Schottky. A current that drops 1v on a 1 amp Schottky might drop only 0.3v on a Schottky rated for 20 amps.
Schottky diodes DO have higher reverse leakage current, potentially much higher. The magnitude increases with the device's current rating, temperature, and reverse voltage. If it is a battery charger for example, this could drain battery current backwards into the transformer secondary winding in a significant way. It could drain the battery when the device has been turned off.