The 25-06 is a much better round. It has a ballistic coefficient, in (almost) all bullets, of greater than .300. Typical velocities with bullets up to 115 grains are over 2,800 fps. Bullet choices range between 87 and 120 grains; and it's possible to push a 120 grain bullet at up to 3071 fps. The range is greater; and so is the accuracy.
As for the 7.62 x 39mm? It's essentially a military round that's designed for primary use in shorter barreled military carbines (AK = 16 inch, and SKS = 20 inch). There's basically one bullet: a (nominal) 123 grain FMJ, or commercial SP. The commercial ballistic coefficient is .283; and the military round comes in somewhere around .270. Velocities average about 2,300 fps out of shorter length carbine barrels. (Wolf offers a 153.6 grain commercial SP round that they list as having a muzzle velocity of slightly over 2,100 fps, and a BC of just over .330 using a 20 inch barrel.)
They, all, sound pretty damned loud to me; but I'd give the edge in both recoil and loudness to the 25-06. Is the difference appreciable? I don't think so; but I haven't fired a 25-06 in years. My best guess is that if you can handle a 243 Winchester then either of these cartridges won't be a problem. All this said, my own personal choice for a (long range) deer hunting round would be the 25-06; but, then again, I'm a, 'reach out and touch um' kind of guy.