SSI and PADI are the two largest certification agencies in the world and are both members of the Recreational Scuba Training Council (RSTC). While they are separate agencies, they both adhere to all RSTC standards for their minimum training requirements, and both are internationally recognized. Certifications for any agency (there are more then 85 recognized agencies) will be accepted world wide. Being separate business, they operate differently and have different business models, but they give the same training and certification. So the short answer to what is the difference between the agencies is…. the initials on the card.
No, all agencies recognize all agencies and accept equivalent levels from other agencies to meet their course prerequisites. To take an advanced class you have to hold an open water certification, it does not matter what agency issued it. This holds true all the way through to the professional levels.