Well I am biased of sorts, I have both of them and a 41, and a couple a little bigger than the 44.
For starters though do you load your own??
If so then I would say pick up something you like in the 44 and go merrily along your way. There are loads from mild to wild for it and bullet weights, and styles, enough to make you sick of looking through them. Go with the 44 and pick out something in the 240-260 grain range, along with a powder that will give you a wide range of loads like 2400 or similar. Or if your the experimental type pick up some Unique, some 2400 and some 296 and you will be set for any occasion.
I'm not saying that the .357 isn't capable of hunting with, but with the heaviest bullet weights your simply just getting started into the light weights for the 44. Trust me when I say, pushing a 180gr .357 to top end, you will still not be getting much, if any more performance than with a .240gr 44 running right around 1000 fps. Then figure you can bump that 240 on up into the 1500fps range and you can start to see where it might be a bit more flexible.
The added weight, and diameter, coupled with being able to push them faster gives you more energy to use on the intended game. Add that to being able to load from light target loads up into full blown magnums, and there really isn't a downside.
Like I mentioned above though, I have them both and several others as well so I am somewhat biased. Even so I enjoy shooting them all, and quite a bit at that.