In Theory, a larger screen would work with a Raspberry Pi. I am not sure about an Arduino as I have never used these before. I think with an Arduino you wouldn't need the Adapter Board as the Arduino has plenty of GPIO pins.
Most displays of this type expect data input of 16 or 24 pins. As the Pi doesn't have this many GPIO, the adapter board turns the signal from a few GPIO (serial) to an output of a lot(parallel) using shift registers.
Mark --OzzMaker.com --