Foundations are to have a bit more substance than classics, but not quite as refined.
Classics are refined, have a more level trot to the ground, do not raise their knees to above level of the shoulder. A foundation has a bit more substance.
Modern Pleasures are a division between the Moderns and the Classics. They have natural movement, their knees naturally raise just below level. They are to have a "pleasure" about them. They look a lot like Moderns, but some are pretty headed and have nice bodies.
Moderns are the oldest division of the ASPC and they have quite a bit of motion, often reaching high above level with their knees. They have a bit more of a spirited look about them, although they are very intelligent and have a work ethic that is hard to compare to any other breed. There was Hackney influence through breeding, but the books have been closed for quite some time now.
If you go back and look at the old Journals and pony magazines from the '50's '60's and early '70's you will not see much difference between the divisions, because there was only one division and all ponies were shown in what we consider the Modern division these days.
Each division is unique, or at least should be. The ASPC pony is a wonder. I bought my first 7 years ago. I was going to do just miniatures, but after seeing a couple mares at a sale - they came home with me. I haven't had so much fun, or so much challenge. They are smart and quick to learn their jobs. They give their all for you, and they want nothing less from you.
I have each division accept Foundation. I started into Hackney's as well and I have a breeding program for ASPR ponies, which is the division that came after the Modern division for the high stepping, fascinating ponies. They are truly fun to work with.