Band saw for speed, scroll saw for precision. That's the simple version.
Personally I've always preferred a band saw, but either will do very nicely. If you're using a scroll saw, I strongly advise using a sharp blade and swapping when it starts to dull. You'll also go through more blades if you're cutting thicker materials on a scroll saw.
Lexan/polycarbonate is certainly possible to cut on a scroll saw, and so is garolite/Micarta. You can also just use tin snips to rough out your bills and then smooth the edges by hand or with a dremel. Personal preference; I usually did it this way because I had limited time available on my brother's scroll saw.