Recoil is as variable as all of the different loads you can make for these two calibers. You can make either one kick like a mule or you can have either one pop like a .22. They are both that versatile in that you have a great extreme with recoil. I have both and have all the different loads for both. I have what I consider to be my plinking/target/varmint loads and my hot hunting/protection loads. You can shoot different levels of 38s in the 357 and will have minimal recoil. You can shoot 44 specials in the 44 mag and do the same thing. You can shoot hot 180-200 grain bullet loads in the 357 and hot 300-325 grain loads in the 44 mag.
To make the right choice, you have to decide what you want the gun for and how much you want to pay to do that. 38s and 357mag ammo will likely be cheaper than 44mag. 38s and 357s are more popular and thus more available.
If you reload, the difference in cost is in the bullets and the amount of powder. Dies, presses, etc would cost the same for either.
For small critters and game up to deer, the 357 is all that you need.
If you want to go for larger whitetails, bear, sasquatch, the 44 would be the better choice.
My newer 44 with round trigger guard and 4 5/8 barrel is a sweetheart to carry and shoot. My 357 has about equal carry time. Both suit me well. You just need to determine if what you are shooting at needs to be dead or DEAD!!
Steve
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