The Model 66 is a "K" frame gun, and the Model 686 is a "L" frame gun.
The 66 is simply a stainless steel Model 19, which was developed at the insistence of Border Patrolmen and legendary lawman Bill Jordan.
Jordan wanted a lighter weight .357 Magnum than the heavy S&W "N" frame revolver for law enforcement.
S&W obliged with what became THE police revolver of model times, the Model 19/66.
In the 1950's and '60's, most police departments practiced with standard or even light-load .38 Special ammo, and loaded up with .357 Magnum for duty use.
In the 70's, most police departments began to use ALL .357 ammo for practice AND duty use, and that turned up a potential problem with the Magnum "K" frame revolvers.
The increased use of full-power Magnum ammo caused increased wear on the guns, and they needed to be repaired sooner.
S&W soon brought out a beefed-up revolver with a new, heavier frame called the "L" frame.
This gun basically copied the lugged barrel and cylinder size from the Colt Python, had a sturdier, heavier frame, but retained the same size grip as the "K" frame.
The new, improved revolver does stand up better to increased Magnum shooting, while being only slightly larger and heavier than the "K" frame Model 19/66, but offers the same grip size to fit average sized hands.
So, if you plan on shooting large quantities of hot Magnum ammo, especially the 110 to 125 grain screamers, you might want to buy a 686.
If you plan to shoot it with mostly .38 Special or .38 Special +P ammo, OR you don't plan on firing many, many thousands of rounds of Magnum, you might want the 66.
If this is a gun you plan on carrying concealed, you'd probably want the slightly smaller, lighter, more compact Model 66.
If you're using this as a range gun, a house gun, or a belt holster gun, get the Model 686, since in those cases, size really isn't an issue.
So, although the grip is the same on both guns, and grips are interchanger able, the Model 66 is a lighter more compact concealment gun, while the 686 is a larger, heavier gun with a much larger diameter cylinder, which makes concealment more difficult.