The software is quite dramatically different which could potentially mean that Picade PCB has slightly more input latency than Picade HAT but I haven’t actually put this to the test.
Picade PCB can be reconfigured to emulate a gamepad, keyboard or weird combination of the two. This is handy if you’re using it with things that don’t accept keyboard input (rare, but sometimes an issue with analog controls) or if you might use it with something other than a Pi.
You can also use two Picade PCBs in gamepad mode together (or more, though I’ve not heard of any more than two being used in any build yet) which is handy if you want extra buttons or player 2 controls. With the HAT you only get the one set of inputs, although you can still add a PCB to that anyway.
You wont lose the 3.5mm headphone jack if you opt to keep analog audio enabled and forego the amp/speaker driver on the Picade HAT itself. Actually the only reason for disabling the analog audio is because RetroPie wont let you select digital out otherwise, and has to be forced to use it. If you’re using another OS then this may not be a problem at all.
I think those are the key differences, anyway.