What is a Miter Box and Saw?The miter box is not really a box in the true sense that has four sides and a top and bottom. A miter box is more like a tray that helps you hold wood in place while you make angled cuts with the miter saw. In addition, this "tray" has slits on the sides that allow you to fit the saw in, and grooves on the bottom that allow you to cut all the way through the wood.
The miter saw, too, is not just an excuse for tool manufacturers to sell you another tool. The miter saw is short, fine-toothed, straight, and typically has extra support along the top to keep the saw blade from buckling. Some products--including the inexpensive Stanley miter box and saw combination, come with clamps to assist you in holding your trim pieces tight while cutting.
What Would You Use a Manual Miter Box and Saw For?Not for cutting two-by-fours, that's for sure. The manual miter boxes and saws are more for fine cuts on millwork (crown molding, quarter-round, etc.). The most common cut is a forty-five degree angle to allow two pieces of wood to join at right angles. Some miter boxes split the angle even further and help you make a 22 1/2 degree cut, but it's unlikely you will use that angle often. (Buy Direct - Stanley Deluxe Miter Box With Saw for making 22.5 degree cuts) Also, all miter boxes will cut ninety degree angles. But that's just straight across, right? You'd be surprised how often you use that miter box for ninety degree cuts. The box simply provides you with a bit more stability even when making those easy cuts.
But when it comes to oddly shaped pieces like crown molding, it is essential to get a claming miter box. Lacking a clamping box, I once tried to cut crown with a regular miter box--to disastrous results. The problem is that one side of the crown has to be set up against the side of the miter box. Ad hoc clamps and supports simply will not work. Your cuts will comically off-angle. The clamping box is what you need. (Buy Direct - Stanley Tools Clamp Miter Box - 20-112).
These Stanley boxes are the staple of many workshops. Made of plastic, their saw slits will eventually get gnawed away by the relentless pressure of the miter saw. But this is just a fact of life with miter boxes. In fact, all miter boxes are built so that they give way before your saw does. My view on the whole thing: buy the cheapest darn miter box available because you'll end up tossing it in the recycling bin after six months anyway.