The model 37 is just a lightweight version of the Model 36 (or Model 60). About 10 or 12 years ago, S&W came up with the brilliant idea to make those guns even lighter by installing a titanium cylinder, and by so doing saving a few more ounces. That's what the first "3" is on the 337. It designates the cylinder (and later the 3 piece barrel) as lighter.
One big thing a buyer can do is to weigh his various J frame grips and find the lightest set he can buy (or install out of his collection of grips.)
I'm guessing the 337 is maybe 3 ounces lighter than the 37. And maybe in the range of half the weight of a M36 or M60. Its not the most pleasant revolver to take to the range for an afternoon of shooting. But they didn't build it to do that. They built it as a very pleasant gun to carry all day, every day, and maybe shoot once in a while.