The current rating is the maximum current that the device will provide. If your device requires a higher current than the device will provide, it will likely overheat the power supply and may also create unusual circumstances within the electronic device.
Consider as an analogy that amperes is water flow. Your device requires a certain amount of water to be flowing in order to work. If the opening in the dam is insufficient to provide this amount of flow, it will still flow, but will not provide the necessary energy to the device to work properly.
If the power supply is rated higher than that which is required by your device, no harm is expected. The power supply will only provide that which is required, and not overheat, and perhaps will survive longer.
Update/edit: Website handheldlegend.com has apparently performed a test on the GameBoy Color to determine power consumption/current draw at maximum volume. It appears that the site is promoting a lower power draw LCD panel, but that does not invalidate the results, table below from the linked site:
One can note from the table that the current draw increases with battery voltage depletion. This is consistent with physics of electrical power. The indicated power supply is likely to provide consistent voltage up to the maximum rated current, allowing one to use only the top line of the table as the reference.
The indicated current draw for that line is below the 300 mAh capacity of the less-expensive power supply, allowing one to avoid purchasing the top of the line model.