The LAA permit is generally for homebuilt aircraft although there are exceptions to this such as: Jodel, Auster, etc.. Aircraft operated on this type of permit are technically restricted to day VFR and should not be flown over towns and villages. There are also some problems with flying to Europe from the UK. Usually people operate these type of aircraft to save money by doing some of the maintenance themselves.
Most other light aircraft now come under the auspices of EASA; which is still in a state of flux with the exception of some types which are Appendix ll. The EASA aircraft operate on an ARC (Annual Review Certificate) and will need to be maintained in a more 'controlled environment'. The advantages are that you can fly at night and IMC (with appropriate licence) etc..
This just a brief overview - hope it helps!