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A power miter saw is basically a circular saw on steroids. It is installed on a hinged frame that allows you to make precise, repetitive vertical cuts either straight or on angle in small boards, lumber, and molding. Repetitive is the key, making the power miter saw a real timesaver!
A compound miter saw not only cuts on an angle, but also allows you to tip the blade to the side, giving you essentially two angled cuts at the same time. This quality makes the compound miter saw a huge advantage in the cutting of moldings... most notoriously crown moldings.
With a standard miter saw, the crown molding must be positioned on the saw table UPSIDE-DOWN at the precise angle at which it is going to be installed before it can be cut at an angle! This is unwieldy at best! To make matters worse, tall crown molding will not fit beneath the blade of a standard miter saw. And even if it did, the raised guide along the rear of the saw frame is usually too short to lean the molding against. This requires you to fabricate a support or angled block to rest the molding against. All in all a do-able but rigorous test of your patience and planning skills. Making and installing the guide or "jig" can take more time than the rest of the cutting!
With a compound miter saw these problems are a thing of the past. The angle of the blade can be set so that you can make the compound cut with the molding lying flat on the saw's table... a vastly easier and more accurate cutting method. Wood Magazine, at http://www.woodmagazine.com, has a nifty software program at their site that helps you compute the compound miter angle for figures with up to 50 sides… a little overkill for a room but invaluable for builders of multisided wood figures such as large birdhouses and gazebos!
When you shop, you will find two types of compound miter saws… the standard compound miter saw and the "slide" compound miter saw. The slide compound miter saw does not merely hinge to make the cut, but either hinges-and-slides or just slides. The advantage is that much thicker and wider boards can be cut. This is the clear choice for the pro, since it offers unsurpassed accuracy in cutting angles on most commonly sized lumber, boards AND moldings.
Compound miter saws tend to be quite a bit more expensive than standard miter saws, and compound slide miter saws even more so. If you plan to do lots of moldings or are willing to spend a little more for the best tool, it will be worth your investment.
As far as the best way to attach crown moldings to the wall, finishing nails are best for wood moldings of all types. Screws are difficult to countersink, especially when working with crowns. They tend to be thin and the screws can either crack them or push right through! There are now some plastic moldings that look like wood when painted, and these are light enough that gluing with a latex adhesive caulk is recommended, but a few finishing nails are still needed to hold them in place until the caulk dries.