What are the advantages of a two-day braise?
Always let a braise cool in its liquid to prevent drying out. As the meat cools, the fibers relax, allowing the juices to be pulled back into the meat. Imagine squeezing out a sponge (heating the meat), and then releasing the sponge in water (cooling the meat). (People on this forum have disagreed with this statement before, but I stand by it.)

Although allowing the meat to cool for 30 minutes or so will yield fantastic results, if you are an overachiever, putting your meat in the refrigerator overnight will increase the effect. After the meat is cold, it will become more solid, making it easier to remove bones, slice the meat, and skim the fat. 

However, I never do this because I am impatient, I kind of like the mouth feel of a little bit of fat, and the results are very good without it. I suggest leaving the refrigerator for the leftovers, which will be wonderful.

BTW, pay less attention to the time, and more attention to the state of the meat. 4 hours may be too much, or two little. The best judge of whether the meat is ready or not, is to taste it. This points to one more advantage of the overnight refrigeration, controlling exactly when you want your food to be ready the next day.