The .357 works very well, provided you use the right bullet and load. As noted earlier, the typical .357 load is intended for handguns. 90% of these FAIL in carbines and at carbine velocities. Unfortunately, there aren't tons of great 357 carbine loads on the market. Its more a handloaders proposition.
The .44 is better. It works great, and its more flexible. Many 44 loads are still intended as handgun loads, and when pushed faster they do tend to come apart. However, the remaining 'pieces' tend to be bigger and heavier and they still manage to punch deeper and get more work done. I've pushed a regular cup and core non-bonded 240 JHP out of a 20" Marlin 1894, and centered a modest woodchuck. We are talking a critter that weighs 10 lbs, and is maybe 8 inches thick. I found the jacket, completely separated from the core, hung up on fur on the exit wound. This is a great example of the wrong bullet - JHP's open too fast at carbine speeds.
Use a 240-300 Jacketed Soft Point and whitetail are in the freezer. There are a fair number of .44 carbines, made and intended for whitetail. Ruger .44 semis and levers. .44 levers from Rossi, Marlin, Winchester and others.
I can get 1900 fps out of a 357 with a 158 Remington JSP over Lil'gun. I can reasonably close to -1800fps - that with other powders. The same rifle, chambered for .44 can push a 240gr to 1700-1800 without an issue. The 240 at 1800 is going to hit far harder than the 158 at 1800.
The .44's are bigger and blunter. They aren't going to suddenly become a longer range rifle. However, within the same 125 yard range, the .44's are going to hit with a lot more authority. And those bigger bullets will tend to drive deeper and harder. Something like Hornady's 265 softpoint would be a real thumper in .44 mag. The real advantage with .44 will be the ability to use a broader selection of factory ammo. Since there are a lot more ,44 carbines out there, factories make more ammo that is suitable for carbines. Off the top of my head I think PMC, Magtec, Win, S&B, Prvi Partizan, Federal and others all make 240 to 300 Jacketed SOFT point ammo. Any of these will work. Under no circumstances would I use the 180 JHP loads. I used these on snowshoe rabbits once and it got.......... messy.
One off topic comment: I know that in theory you can use .44 Special loads in a .44 Magnum carbine. Yes, they do load and fire. Forget about the concept. its not practical. I saw a good six inches of point of impact change at ranges of less than 25 yards with the two loads.The difference in trajectories and point of impact make switching between these two an idea that simply does not work well.