Well, I just got a EU driver's licence (replacing the old one that had my 18 year old face:-(;
Going to the States, I'd show my passport whenever you'd show your driver's licence.
Now, let's suppose the police stops me in the car and asks for the papers (insurance of the car, my driver's licence and ID card; yes, I'd better have all of them all the time, when I'm driving). If I didn't have one of those papers, they'd issue me the order to present all those papers at the closest police office , the next morning, but would let me free. If I were walking without my ID, they'd also ask me to present that paper asap (after having taken my name and address).
Raoul