Piston pens have the filling system fully contained. The only place ink can come out is the nib. Converters, in my limited experience, don't always have a perfect fit or come loose during the filling process.

More deft hands probably don't have this problem. I would come out with inky hands no matter the filling system when I first started using fountain pens. Look ma', no (stained) hands!

They’re sort of similar, but also opposite. A convertor twists up the pen pulling ink into the chamber. A piston twists down the pen creating a vacuum first. When it gets to the bottom the vacuum is exposed to the nib and the ink is rapidly sucked up. I definitely find the piston less messy.