I doubt that you or the game intended will notice the difference most if the time. If you are still shopping, it is easier to get factory ammo in .44 and the ability to shoot .44 special for non hunting use adds flexibility.
In my opinion, full power 44 vs full power 41, the 44 recoils a bit more.
But if you handload, that is a non issue.
There are a lot of 41 fans that are hoping Ruger/ Marlin makes an 1894 in 41M.
The difference is likely negligible...but a .41 gun will weigh somewhat more than a .44 in the same frame revolver as there's more steel left in the barrel.
I like this one for hunting using stout .41 magnum ammo. I load up 265 grain hard cast LSWC ammo.
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This is a 657-2 Classic Hunter and it seems to recoil a bit less than my plain-jane 6" Model 29-3 .44 Magnum with similar loads.