Not long ago there was a good deal of discussion on the roof prism issue, and some interesting papers were discussed on this forum. Here are some things I learned, hey I'm not smart, this is just a great place to hang out:
Schmidt-Pechans make a very compact binocular, because the light bounces all around in them and the optical path gets strongly "folded", in just a little space of length. Abbe-Koenigs are long and skinny, and the light goes more nearly straight through them. That is why some of the older pre-fluorite Zeiss Classics are so long.
Longer focal length is good, optically, because it decreases aberrations like color and spherical aberration. If an AK bino is as compact as a SP bino, it has a shorter focal length. To get back the image quality, some other trick must be pulled, such as the use of Fluorite in the objective of the Zeiss FL.
All the reflections in the AK are total internal, not requiring coatings for perfect reflection. One reflection in the SP requires a coating, and this being of aluminum or silver is the main reason that SPs of the past were dimmer. Now, multilayer dielectrics even that up, but you still have to wonder if complexity can ever equal simplicity, in perfection.
Also, one of the surfaces of the SP does double duty, both admitting light from the outside, and also providing internal reflection on the inside. The use of the best antireflection coating on this surface, to optimize the admission of light into the prism, unfortunately compromises the internal reflection. So the coating on that surface is often compromised, to give the best overall image.
One would hope that with today's miraculous technologies, the practical differences between the designs are extremely subtle to nonexistent, but they've been saying that since Galileo.
I am trying to talk myself into an FL here, can you tell?
Ron