The problem with the M19 is that it doesn't hold up as well with .357 magnum loads. Forcing cone erosion was a major problem.
When the M19 was introduced, conventional training wisdom by law enforcement was to carry .357's but to train with .38's as "practice ammo" either the traditional 158 gr lead roundnose, or the 146 gr target wadcutter, saving the .357 for duty. Police training changed and .357 ammo improved from the standard 158 gr semi-wadcutter to 125gr or 110gr jacketed hollowpoints. This is when it became apparent that the M19 couldn't hold up. The 586/686 (581/681 in fixed sight versions) came out and was S&W's solution to the problem. In terms of frame dimensions, they were very similar to both the Colt Python and Trooper Series (which have similar frame sizes "I" frame, I believe but much different lockworks). About the same time, Ruger came out with the GP100 series to address the same problem, although by all accounts the Security Six revolvers were pretty durable.
A lot of people love the Model 19 (one of dwarven1's favorite handguns, I believe). It has been characterized as a "carry much, shoot little" gun. I have always thought of it as a very durable .38 Spl with a .357 capability on occasion.
So as others have said: what do you want to use your revolver for? If you shoot mainly .38s, then go with the M19, if you are going to shoot a lot of .357 then go with the 586. I have a personal preference for the 586/686 series myself and have owned both the stainless and blue models.