Well, you wouldn't if you didn't want to fly a non-US registered aircraft.
You can for example with a non-JAA ICAO License (e.g. an FAA one) fly a G reg aircraft in UK AIRSPACE. But you could not fly it in the airspace of another JAA member state. Nor can you fly it within the UK within CAS where compliance with IFR is required, nor can you fly it to give instruction in flying.
Myth....
I've got in confirmed in writing from the DGAC (French) and the Germans that I can fly a G reg using an FAA certificate in their airspace.