I did this route, although it was a few years back (1990s).
Firstly, please only take advice from those who have flown in both USA and UK/Europe. I have heard so many pilots who have only flown in the UK say that it is the best/only place to learn to fly. Ask most people who have flown in the USA as well and you will get a much more balanced view. So best to speak to those who have done both FAA and JAA licences.
I did FAA PPL/ME/Single engine IR/CPL (all part-time, whilst on planned "vacation" trips) and then I gradually converted to CAA frozen ATPL (now JAA!). Shortly (actually very shortly!) after which I stared flying a jet for a UK airline.
I had to do the UK ATPL ground exams (although I think I was exempt from taking Morse because I had an FAA licence. Even though you don't do morse for that either...), the PPL and Multi were a straight swap to JAA, apart from doing a couple of exams I think (RT and Human Factors I think). I had to do "sufficient" training for the UK ME IR (did 10 hours including test and positioning to the airfield to start the test) and then I did a flight test to get my first commercial rating - on a C172!!!
I've flown a fair amount now in the USA, as I did some hour building there, and flew from California to Florida (Key West) and back, and also from San Diego up to Vancouver and back, as well as various other flying out there.
I also flew a light twin from the UK around Europe.
I would say:-
Benefits of US - flying is so much easier, you can do lots of it, really get into a routine, visit lots of airports and get familiar with structured flying into big airports (the sort of thing you will be doing if you go onto the airlines). You can fly at night, its no big issue. There are some great places to go and see, and when you land you can often get a free car and accommodation. The weather can vary hugely (whilst doing early hour building I had serious IMC weather, some very interesting winds, all sorts. I had way more exposure to different weather conditions in the USA than I ever did in Europe). Instrument approaches are free, so you can practice to your heart's delight, and there are both procedural airfields and radar airfields. You generally have to think on your feet a bit more, but the back up is very good if you need something.
Benefits of UK - radio is more "proper" phrases. There are NDBs. IR flying is much more pre-planned and "booked" than in the USA. (Often in the US you can just get airborne from a non-controlled field, and call up on the radio to file your IR flight plan). You can pay plenty of landing fees and approach fees. It is a lot more "procedural" when flying light aircraft at small fields (not so much when flying airliners as at the big airports its usually radar vectored to ILS).
Personally, having done a few hundred hours light aircraft flying in each place, I would much rather go to the US. The kind of flying I did out there was much more aligned to airline flying (eg. file IFR flight plan, get clearance, take off from medium sized airport in San Diego, radar monitored, fly up to Las Vegas McCarran Intl, vectored for ILS, land, get taxi instructions, arrive at handling agent. Park, get transport back to office.). No charges with regards landings or routes, parking charge may apply (usually not if you take fuel).