On a most basic level the styles of pilates can be broken down into Classical and Contemporary.
STOTT is one type of contemporary pilates. Full disclosure: I’m certified in STOTT. This does not mean that I think they are always right (in my classes I incorporate things I have learned from the elders and other programs). I do think that if you have a good Instructor Trainer when you go through STOTT’s certification program, you’ll get a very good education. I had a wonderful trainer in my program and I continue to train with her.
STOTT, like other types of contemporary pilates, has taken the traditional pilates exercises and modified many of them. There is still the option to do the original exercise, but the modifications provide more options for more people. The breath is very focused and in time with the movement. STOTT (as with other contemporary programs) incorporates the neutral pelvis. This differs from classical pilates where every exercises is done with a flat, back, what is now referred to as an imprinted spine.
Of contemporary styles of pialtes, I do think STOTT is one of the best. They have modified the exercises based on research and work with physical therapists to determine what is best for the body.
Even in classical pilates you will find different styles. Jay Grimes, Lolita san Miguel, Romana, and Mari Windsor would be some of the people that are probably teaching close to classical/authentic pilates.
In Classical pilates the spine is imprinted. The breathing can be different and even reversed from some of the contemporary breath. In the case of Jay Grimes, he claims that on most exercises you should just breathe normally as if you were walking down the street.
Classical pilates usually has the legs laterally rotated (turned out at the hips like you are Mary Poppins) as opposed to parallel legs like contemporary pilates.
Even within classical pilates there are some off shoots.
Fletcher Pilates really focuses on the breath. And has developed towel work to go along with the pilates exercises. His teaching is very dance-like (which was not typical of exercises Joe Pilates taught).
Whether you are taking Contemporary or Classical pilates, it’s really all about the instructor and you. Each instructor brings their own personality, experience, and interpretation of an exercise. No one is Joe Pilates, but we are all trying to emulate a piece of him and his work.