From an aerodynamic standpoint, they both do the same thing.
From a conceptual standpoint, I'd say a fixed slat is something separate bolted onto a given airfoil, whereas a slotted wing is an air channel cut within that airfoil.
In other words, take a given airfoil, and bolt something on the front of it to produce a slot, and that is a fixed slat. Take it off, and you are back to your original airfoil.
Conversely, take a given airfoil and cut a slot through it, and the overall external profile of the airfoil is still the same. Remove the slot (and the portion of airfoil ahead of it) and you no longer have the original airfoil.