Interesting question.
In Finland, neither is very commonplace as hunting caliber although of course since the 7,62x39 is the military caliber, it's common enough.
I know of some who use the 7,62 for bird hunting, but it's not very flat shooting and thus not the best suited for the purpose. The same goes for the old 7x33mm Sako which is now very rare.
More often the .308 or 6,5x55 is used.
For distances over 100m I see no benefit in choosing the 7,62mm.
I'm of the same opinion even if a bit bigger game is to be shot. However, if you intend to shoot game bigger than whitetail deer, I'd recommend a bigger caliber.
The .243 is very flat shooting and accurate, indeed. The recoil is surprisingly sharp, though.
Come to think of it, I really don't see much benefit at all in any hunting to use the 7,62x39. If you wish to have a caliber with good ammo availability and cheap ammo, why not look at the flexible 7,62x53R? That should be common enough in Ukraine!
What do you intend to hunt?
I'd be interested in hunting in Ukraine, what do you typically hunt there? I know that in the Western parts where there are forests and high hills, big game like wild boar is available.

- Lars/Finland