PADI, the Professional Association of Diving Instructors was founded in 1966. The agency has been around a long time and is still the biggest - their name and logo are universally recognised and is synonymous with SCUBA diving. Their modular approach to diver training has been copied and adapted by every other training agency.
Like all the major dive training agencies you will never have any issues having your qualification recognised anywhere in the world you choose to dive. Some agencies have come and gone; some agencies are new and not as widespread as they'd have you believe. No such issue with PADI.
Does PADI offer training that is any "better" than the others. Read their blurbs! Every agency including PADI would have you believing they had reinvented the dive course and their courses are "the best". Lots of shops will tell you the same thing. I've taught through a number of agencies: they all teach to the same basic standards: they all allow some flexibility in their courses to allow for training in different environments around the world and they all have minimum standards. What makes a quality dive course is not the agency used but whether the instructor goes beyond minimum standards - often it's an issue with shop economics not the instructor; that time constraints don't reduce a student's ability to gain confidence and that class numbers are small enough that students don't spend all their time waiting for others to complete skills. PADI is still the benchmark everyone else compares themselves to: PADI allows us at GNC the flexibility to teach the best dive courses possible.
The REAL difference between PADI and all the other agencies when you get to the professional level. Once you are a PADI Divemaster or a PADI Instructor you have that qualification and can work anywhere to that qualification. With other agencies you can ONLY use that qualification if there is a local shop willing to employ you, there is no way you can work independently. It is easier to cross-over from PADI instructor to any other agency. The PADI IDC (instructor Development Course) is the most onerous instructor course there is in Recreational Diving. A short cross-over and you can teach through any other agency.