The 4 amp and 15 amp ratings refer to the power supply's maximum capacity available. Just because it's rated at 15 amps doesn't mean it'll cook anything (unless you connect it incorrectly). It just means that it has more than enough power available to use if needed by the gameboard. in the old video game days power supplies didn't have much power capacity (amps) available. As time went on power supply manufacturers built power supplies with a higher amperage rating so that they could make one power supply to cover a greater number of applications.
Think of it like two car batteries, just one is bigger than the other. Sure they'll both start your car, but the bigger one will give better performance.