I have little understanding in regards to the differences between the two, although I do have a few books on classical dressage which often dismiss modern dressage. From what I have gathered the differences are about how force full and trainer is with a horse. Training a horse in the classical manner will take allot longer than the modern methods, so therefore it yield quick results
I also understand that a horse that has been classically trained will never be able to succeed at the top because it hasn't been forcefully trained to do 'flashy movements' to impress the 'ignorant spectators who want circus like shows'. In other words, the modern dressage horse does not represent truly natural movements like a classically trained horse, but will always succeed in competition for pure entertainment reasons.
As I have already said, these are not my own interpretations of classical and modern dressage.