No difference.
The Miata is badged MX-5 Miata in the U.S. and Canada. In Europe, it's the Mazda MX-5. In Japan, it'sthe Mazda Roadster, and it used to be sold under Mazda's Eunos brand, as the Eunos Roadster.
The '04 Miatas in the U.S. are still called MX-5 Miatas, except for the factory turbo model which is the Mazdaspeed MX-5. Rumor has it that next year they will no longer be called Miatas here at all, and will all be MX-5's.
They're great cars, and they last forever. They're also reasonably safe. If your son is a new driver, make sure he has some training or experience driving a rear wheel drive car in the wet before you unleash him on the streets. I'd suggest having him autocross it on a rainy day. I learned more about RWD and skid control in one day of that than I did in years of driving.