I like the setup at Crystal River in Florida - small, professional, with highly experienced people. The owner flew from WW2 carriers with George Bush (senior). http://www.crystalaerogroup.com/introduction.html
A JAA PPL allows you to fly any JAA registered aircraft anywhere in the world.
An FAA PPL allows you to fly an 'N' registered aircraft anywhere in the world.
The confusion arises when you mix the 2. Many countries, Ireland included, allow PPL priviliges (VFR ONLY) on 'their' aircraft within 'their' airspace with ICAO compliant licences. So, for instance, my understanding is that you can fly (privately, not for remuneration) any Irish-registered aircraft type permitted on your FAA licence under VFR in Ireland using your FAA licence.
Some (eg the USA and South Africa) require you to 'validate' your 'foreign' ICAO PPL to a national licence. For example in the USA the so-called 'FAA piggyback' PPL is issued on the strength of a 'foreign' ICAO-compliant licence. The FAA licence is instantly invalidated if the 'parent' licence expires for whatever reason.
Both JAA and FAA licences are recognised around the world, each has pros and cons. An often quoted 'advantage' of an FAA licence is that it is easier for an FAA PPL holder to obtain and keep an instrument rating than it is in the JAA system. This is probably one of the reasons why so many private FAA-registered aircraft are used in Europe.