Will the Light Aircraft Pilot License (LAPL) or Private Pilot License (PPL) be enough to to be able to fly to other countries outside Europe?
The EASA issued LAPL(A) is limited to 2to MTOM, 3 passengers max and only valid in European countries. A PPL(A) issued by an EASA country does not have these restrictions, but can be subject to local restrictions in the non-European country you want to flyin, e.g. you cannot use the EASA PPL(A) to fly in the US without further paperwork.
For more information, see the UK CAA's page on the LAPL.
Your LAPL(A) is a lifetime licence that contains SEP(land) and/or TMG privileges restricted to non-complex EASA aeroplanes with a maximum take off mass of 2 metric tonnes and a maximum of four people on board.
You can fly more complex aeroplanes within the privileges if you have 'differences training' signed off in your log book by a suitable instructor. The LAPL is also valid on non-EASA UK registered aircraft in the UK.
The LAPL requires an in-date EASA Part-Med Medical Certificate, at LAPL level, but Class One or Class Two certificates can also be used. A valid JAA Class One or Two medical certificate is also acceptable. But you cannot use your LAPL with an NPPL medical declaration.
The SEP(land) or TMG endorsement on this type of licence is not issued as a rating. So there is no rating expiry date. Instead you must follow self-validity checks on your own flying history for the past 2 years to ensure currency on the day of your planned flight.
Further Reading from the UK CAA:
EASA Light Aircraft Pilot License Aeroplane - LAPL(A)
EASA Private Pilots Licence Aeroplane - PPL(A)