I have come across a curious problem. The Catalyst fp model is approved for both NSA and ASA. The Catalyst sp bat is not approved for ASA but it is approved for NSA. I had a player in a NSA fp tournament this weekend come to the plate with a sp model Catalyst. I do not think there is a problem with this but it did make me curious.

What exactly are the differences between sp and fp softball bats?

Weight, and marketing claims. The official bat, according to ASA, may not exceed 34" in length, nor 38 ounces in weight. Generally, "fastpitch" bats are -11 to maybe -7, meaning the 34" versions weigh 23 to 27 ounces. Bats heavier than that are marketed as slow pitch bats, whose players have all the time in the world to get that tree turned around.

I think I know the answer to this, but for everyone's benefit...
 

Can a bat marked "Fastpitch" be used in the slowpitch game (and vice versa?)

Yes, As long as it is a legal bat then it can be used in either.