The slight difference in length is just a safety precaution, nothing to do with power. Its there to simply prevent .357 magnum ammunition going into .38 Spl. chambers which would be a disaster. The .38 Spl. is actually a black powder round originally so the case is huge for most modern smokless powders because of the large difference in energy available per grain burnt.
The real difference is internal, & not easily visible, its the type & volume of powder used & the pressure levels it works at. As an example a .38 Spl runs about as hot as is sane with 4.8 grains (a grain is 1/7,000 Lb, so thats a very small amount) of powder & pushes the bullet at 930 Feet per second, it generates about 15,500 CUP of pressure. Now a .357 Magnum can go as high as 22.0 grains of powder pushing the exact same bullet at 1,966 FPS at a pressure of 41,400 CUP of pressure.
Different types of powders are used as well, essentially fast (for rifles) powders are considered "slow" in a pistol & are harder to ignite reliably, so the "magnum Primer" function is just to give a bigger spark to get the slower powderrs such as H-110, or 2400 burning. If you use powders like "Titegroup" for example you can safely run standard pistol primers & still get higher velocities from .357 loads.
As you can imagine the difference in recoil will be considerable because of every action having an equal & opposite reaction! If you ran the .357 Mag load's 41,400 of pressure in a gun designed to handle 15,500 the result would be catastrophic, so the safety of making it impossible to make that mistake is needed!