Stainless steel isn't as subject to erosion of the throat just in front of the chamber, and the bore itself isn't as prone to rust.
Some people believe that stainless bores don't foul as badly from copper jackets, and are easier to clean.
A heavy barrel heats up slower and is stiffer, so it flexes less.
A carbon steel barrel really doesn't have any real advantages over stainless.
A thinner barrel is lighter and is easier to carry, particularly with a rifle that has to be carried long distances, especially in mountains, and is usually just as accurate as a heavy barrel for the first shot or two.
Thin, light weight barrels are best for hunting rifles that are likely to be fired only a couple of shots.
Heavy barrels are best on varmint, target, or other long range rifles where consistent performance is necessary when the rifle is fired a number of times in a short period of time.