I have no experience of learning to fly in the US but have only one observation, a chap returned to our flying club having gone this route and whilst possessing the licence was woefully prepared for UK weather and traffic conditions and regulation, so much so that he undertook a course of instruction here to enable him to cope !.
That's not reflecting on bad training, that's just a matter of different airspace, rules, weather etc. If you were to do your PPL in the UK and then go to the US to fly there, you'd be woefully underprepared as well. Regardless of whether your license says "JAA" or "FAA".
In both cases it's a very good idea to read up on the rules and regulations in the airspace you're going to fly in, and get a few hours of "differences" training from a local instructor.
And to be honest, we regularly get questions here on UK trained PPLs that want to fly to mainland Europe, or the other way around. That's really not all that much different. Despite ICAO, every country has its own peculiarities and you have to prepare yourself properly if you're going to fly in a different environment. Of course, the more experience you have, the easier it is to take things "in stride".