I'm not familiar with the PADI course, but no one else has been able to give a direct comparison, so I will try to describe SSI's Stress and Rescue and see if anybody can draw a distinction.
1) Role of Instructor and Agency. SSI's general approach is what we refer to as an 80/20 rule. 80% of the course content should follow the SSI materials and 20% is at the discretion of the instructor. You will often read that the biggest difference is between instructors rather than agencies. With the SSI approach, there can be a pretty significant difference.
2) Emphasis on Stress. A big part of the SSI course is devoted to understanding the causes of stress, the physiology of stress and the panic cycle, recognizing the signs of stress in yourself and others, and managing stress in yourself and others. If you can recognize and manage stress, you can reduce the need for a rescue.
3) Rescue Skills. When you look at the academic materials for the course, only one out of the six chapters covers rescue skills. Three cover stress issues, one covers accident management, and one covers factors complicating rescues.
4) Open water sessions. When you go to open water, a greater proportion of the time is actually spent on rescue skills, but the stress recognition, prevention, and managment is also covered, along with accident management.
5) Approach to Skills. The course teaches a simple approach to skills, as DeepSeaDan described. The objective is to make it possible to remember the skills and to make them work in any situation with any gear configuration.
The full name of the SSI course is actually "Diver Stress & Rescue," which distinguishes it from the "Dive Leader Stress & Rescue" component of the training for Dive Control Specialist, SSI's entry-level dive leader training, which combines the roles of divemaster and assistant instructor. When we got to the Dive Leader Stress & Rescue, we covered things in greater depth and a higher degree of mastery was required, but we had already been introduced to the key components in the Diver Stress & Rescue class.
Stress & Rescue was, by far, the most useful course I took before DiveCon. Close second was a course developed by an instructor at our shop called Tranquility Diving. Tranquility covered advanced buoyancy skills, breathing control, proper weighting, and the whole mindset of being still, quiet, and relaxed - in a word, tranquil - in the water. Everybody who took that class dropped a significant amount of lead, improved air consumption, and generally became a much better diver.