Technically it's the amount of prize money the show gives out. According to USEF they have to give away a certain amount to be considered an A or AA show. Because they have more prize money to offer those will be the more competitive shows that cost much more! Your costs for the week (usually they run tues-sun) can easily be in the thousands for the week. You will see a much higher quality of horses and riders with much better turnout. (think, high end custom clothes and professional groomers. Not all, but the bigger shows will). WEF (Winter Equestrian Festival in Wellington FL) is one of the top shows in the country, and they offer cross rails and short stirrup! So height of jumps doesn't matter.
B shows are more local shows. A lot will depend on your area. Our B shows are actually kind of competitive. Pretty nice horses. Most people don't braid but everyone wears coats and very few will wear half chaps. Most people on our B circuit also frequent the nearby A shows. The biggest class our H/J circuit offers is 3", although another circuit in our state offers a 3"6 class. We don't have a cross rails class, but the other circuit does. My friend in another state competes on a circuit that doesn't have anything under 2"3. Ours are usually just 1 days, sometimes 2 depending on the show.
C shows are schooling shows. Again, a lot will depend on your area. Generally they cater to the people who are starting off in showing, or taking their green horses to a place that's cheaper and less competitive just to get them in the ring. The level of competition is usually not as high. I don't know of any schooling shows that are more then a day, but I'm sure they're out there.
I would disagree about the quality of facilities. Some of the bigger shows have terrible stalls and horrid footing. There are several barns at WEF that stay in tents. Some of the HITS shows have terrible footing. I believe Washington stalls in parking garages. I've been to schooling shows with fabulous brand new stalls and nice footing.