A miter saw cuts miters and straight cuts across the grain (crosscuts). The angle of the miter is limited but is usually from 45 (left and right of 90 degrees) and, of course, 90 degrees. This would be good for door and window trims, picture frames, and cutting angles for hexagons.
A compound miter saw will also cut as a miter saw but will additionally cut beveled miters alone or in addition to, say, a 45 degree cut. This is especially handy when cutting crown mouldings or other specialty cuts.
Hope this helps.
Your post was kinda confusing to me. Right now that isn't real hard to do. Go to www.popularwoodworking.com/features/mag/html about half way down the page there is a download for excel. It is really cool. All you do is type in a couple numbers and it gives you the blade tilt and miter guage setting.
A little visualization is necessary to understand a miter cut.
Cutting an inside corner on a crown can be done one of two ways on CMS.
Easiest way - if your crown is not too deep - is to make jig that will allow you to stand the crown up as if it were being attached to the wall/ceiling. Then you simply set the CMS to cut a straight 45d cut with no compounding.
If you crown is too tall - meaning you can not stand it up using a jig - then you need to make a compound cut. I apologize, but I have forgotten the two angles that need to be set. But to show you the problem, set a piece of crown flat on the CMS table and cut two pieces at opposite 45d angles. It will be very evident that you can not fit these two pieces together to make the inside corner, and you'll understand the need to make a compound angled cut. Neither required angle is 45d. The same problem exists for an outside corner.
By the way, "REAL" carpenters don't use a CMS to cut inside corners, the cope them. :D