Both stews and casseroles are ways of cooking meat slowly. The meat is cut up fairly small and cooked in a liquid such as stock, wine, water or cider. This liquid is absorbed by the meat to make it more tender.
They can be left to simmer slowly without attention from the cook. Both consist of meat and vegetables, cooked slowly over a long period of time making it ideal for cheaper cuts of meat. The slower cooking makes the meat more tender, no matter what part of the animal it comes from.
Today the two terms are totally interchangeable, but they are often differentiated by how they are cooked. It is often accepted that a stew is made on the top of a cooker with heat being applied directly to the underneath of the pot, whilst a casserole cooks inside an oven with heat circulating all around the pot.
Cooked with the heat applied from the bottom, stews would have traditionally been cooked in a large pot on the hob or suspended over a open fire. Stewing is a slow method of cooking, especially suitable for tenderising tough meat.
Leftover cuts of meat and vegetables would be continuously added to the pot over several days and allowed to 'stew'. This was often used as a method to ensure that the stew could be stretched out to provide many meals during the week.
Casserole A casserole is the name of the pot used for cooking and slowly has become known as the cooking method as well. Casserole is a large dish with a fitted lid, used for slow cooking. Traditionally a casserole dish is cooked in the oven.
Braising, like casseroling, is done in the oven, but the meat is cooked in much larger pieces and only a minimum of liquid is added, so that the meat actually cooks in the steam for the most part.