Show me your bullets and let me see you shoot both guns before I answer. Shoot a solid, heavy bullet that will exit a deer and shoot it well, in either caliber, and you will be fine. I take shot placement and penetration over caliber, hype and velocity.
The 44 is looked at as being a superior deer hunting round (and probably is). However, let's say that you shoot 3000 rounds a year. How many of those rounds are actually at deer? Most of them will be at paper or tin cans. Sometimes you have to ask yourself which one will you be most likely to practice with the most. Use shooting sticks in the field and stick with the heaviest, solid bullet that you and your gun shoots most accurately.
Which is better 357 or 44? Where did the bullet hit is a better question? I'll take a 357 to the heart of a deer over a shot to the ankle, with a 44, due to flinching.
Which is better 357 or 44? What bullet are you using? I'll take a heavy, accurate, solid, 357 round that penetrates over an expanding 44 that only splashes and doesn't leave an exit (blood drain).
Which is better 357 or 44? If you are looking at cost then I would say 357. You can get good with your trigger in the off season with cheap and light 38's.
Which is better 357 or 44? Do you hunt deer in bear country? If so then maybe the 44 is a better choice.
There are so many variables to consider. I have been in states where deer are about the size of large dogs while in others states they grow to four times that size. You may want some beef behind your hunting rounds if you are hunting larger deer.
I know a hunter who once said, "I'm only going to take 15 yard shots, at deer, from a 20 foot high treestand, with a hollow point "hunting" bullet, out of a 44 mag." Think of that shot angle. He kept running out of blood while tracking the deer that he had shot. If you shoot at close range, from a stand, with a bullet that spashes then you don't have a drain on the low side (or the exit). Who cares what caliber you used if the animal ran xxxx number of yards and died with no blood for you to follow? You know that feeling when you are tracking an animal and you run out of blood? Panic sets in....people start to walk in circles....someone yells out, "maybe he back tracked." Next thing you know a few days have passed by and you have to end the search.
Folks, this situation can be prevented with a bit of planning. I have always heard the phrase, "you can't argue with power when it comes to hunting bullets/calibers." I say....you can't argue with wisdom. Be smart about what you do when it comes to hunting and have fun. Shoot a heavy, solid bullet. Shoot off of a rest when hunting (shooting sticks work well). Invest in quality LED flashlights like Surefire brand or others. Wait at least an hour or two (if not over night) before tracking a deer that has been shot. THESE are some of the things that will lead to success in the field. Caliber selection will take a back seat, to some of the more important aspects of hunting/woodsmanship, as you become more experienced.
Now, with all of that being said, if all things were equal, would I say that a heavy, solid, 44 mag is better than a heavy, solid 357? Yes. I just didn't want to answer the OP's question without talking about some of the other aspect of the equation. I would like to hear what other hunters have learned as well. So, let's keep this dialog going. This is a great place to talk about these types of issues.