The difference is, a fast switching diode will become non conductive a lot sooner than a non-fast switching diode.
In non fast recovery diodes, the path between the anode and cathode is still some what conductive after the current that placed them in that state is gone. What this means is, reverse polarity on this diode will cause a conductive path in the reverse direction for a per scribed amount of time just after a completion of a forward path on the diode if not sufficient time given to have it recover before this reverse path appears across the nodes of the diode.
It acts like a cap that has been charged and when reverse voltage appears across it, it'll exert current. But don't get confused, it's not the same thing. It's all about electrons and the way they impact the surface of the materials between the anode and cathode used in the diode. Fast diodes are made a little differently to limit this problem.
TO get a better understanding, look up Ultra Fast Diodes or Switching diodes..
P.S. Transistors of many kinds are also spec'ed this way.
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