Whenever Smith & Wesson makes a change, or more typically a small set of changes in design and/or engineering, to an in-production gun, they indicate it by starting a new "dash" series for that model number. A "no dash" gun, simply refers to it being the original production configuration for that model number.
A new dash (from -3 to -4, for example) may indicate any number of changes, good or bad or neutral. It could be a change from a forged trigger to a MIM trigger, the addition of an internal lock, the transition from a hammer-mounted firing pin to a frame-mounted firing pin, a different type of steel in a certain part, the type/process of finish, that sort of thing.
A 586 no dash simply means it is the original early-80's configuration: Forged hammer with hammer-mounted firing pin, forged trigger, no internal lock, walnut target grips with S&W medallion, etc. At some point, I believe all or most of those components were changed. This is pretty much the story with most of the Smith and Wesson revolver line. Thus, a new 686-6 would be the "same gun" as a 686 no dash, in that they are the same model number and are the same basic gun, but there are a considerable amount of differences between the two guns. How big or small those differences are is a matter of (much, intense) debate.
In addition, sometimes changes are made that S&W does not indicate with a new "dash". For example, per the earlier conversation, the S&W stamp on the sideplate was removed from the 586 at some point early in production. If my information and memory are correct, there was no dash to designate this specific change.
If you have questions about this stuff, THR is the place to ask. There are some real Smith and Wesson revolver buffs here who seem to know every "dash" and other little detail by heart!