IMO, the main difference between a PADI cavern course and the cavern courses from the major cave diving agencies is the primary purpose of the course. I am in theory a PADI cavern instructor, but living in Colorado with no caverns nearby, I have never taught the course. As I see it, the primary purpose of the PADI cavern course is to make recreational divers safe divers in a typical cavern. That is how I would approach the class.
Respectfully speaking, here is the problem with that approach. The minute something goes wrong, those cavern graduates are in a cave environment.
By definition, there are no caverns without sunlight. An errant fin kick causing a silt out, or some event inside of the cave system (either natural or man made) causing a loss of natural daylight puts divers into that cave environment.
Two or three years ago there was an event inside the cave system that reduced the visibility in the "Ginnie Springs Ballroom" to one or two feet. The ballroom is considered a "safe cavern for openwater divers," but it wasn't on that day.
I have also done deco in caverns that suddenly turned into walls of mud because of the actions of another team further in the system.
And of course we've all heard stories about cavern certified divers that went into the cave zone for just a little peek.
If every training agency gave the environment the respect it is due, and treated a cavern course as step 1 of a cave program, the training would provide the basic tools for survival whether or not the graduate decided to pursue cave diving.
BTW, I've been doing cave diving accident analysis for a long time, and have even published some material on the subject (google it if you like), and I have a pretty good understanding of the concepts, risks, and hazards. And just like it was when the little blue book was first published, training (or lack thereof) is still number one.