I'll take a shot at it.
They go via two different mechanisms completely. Think of it this way, if you're a large molecule and you want to attack an electrophilic species, then you're too big and bulky to be able to hit it head on with any chance of actually reacting(especially since the polar solvent is encapsulating you) therefore you need a highly electrophilic species to attract you.
In an protic solvent, the protons in solvent can turn bad leaving groups into good ones by protonating them. the resulting carbocation is a huge electrophile. A "basic" (and most of the time, small/highly polar) nuceophile would not be able to break through the "capsule" that the solvent makes around a small molecule.
In aprotic solvent, the issue of being encapsulated is gone(since most/all are non-polar) therefore you can more easily attack an unhidered even weakly electophilic species.
This is more or less how I understand it. Hope this helps.
