I love the 44-40 it's areat round "BUT" for hunting you better be a handloader.factory and cowboy loads are real whimpy. my 44-40 is a navy arms win 92 clone and they like marlin are a extremely strong action and you can brew up some great hunting loads.
Shooting the Ruger New Model Super Blackhawk single action .44 magnum revolver. This motivated experiment scientifically pits the magnum up against it's 1/8" smaller "special" brother. Which motivated munition will massacre the melons you ask? Err.
what is the difference between a .44-40 round and the .44 magnum? ... The 44-40 is a slightly bottlenecked case, the 44 magnum is a straight walled case, and it's taller. 44-40 is about 800fps, 44 magnum is about 1500fps for a 225gr lead bullet. I don't believe you compared cartridges if you think they look alike.
I reload both the 44 mag and the 44-40. I shoot the magnum rounds in a Henry and the 44-40 rounds in a Winchester model 92. The 44-40 shells are much thinner, therefor they are not capable of holding as much pressure as the magnum round. Both use the same bullet. The older shell can be squeezed together somewhat with just your fingers/thumb.
If you're shooting factory ammo, the 44/40 would be a lot more expensive. For reduced recoil, stick with the 44 Special loads, or take up the hobby of reloading and you can make your own 44 Magnum loads that are easy on the wrists. Welcome to the forum. #
.44-40 was a completely new round when it was introduced 1873, the same year as the .45 Colt, they are very different rounds. The .44 Henry was a rimfire, underpowered round first used in the original Henry rifle and far outclassed by the more powerful 56-56 Spencer.
As said, it should not be possible to shoot .44 Magnum in a .44-40 and would be dangerous if you could cram one in or ream the chamber to an odd mongrel shape. It is not even wise to shoot hot .44 Magnum loads in a rebarrelled .44-40. The barrel shank is small and the receiver ring is thin for those pressures.