The two rounds are pretty similar. The .308 was designed as a short-action replacement for the .30-06, as new powders became available. The .30-06 is still a bit more versatile, with a larger case, so you can play with loads more. And it can be loaded with heavier bullets. Commercially available high-power .30-06 ammo (Buffalo Bore and Hornady) approaches .300 Win Mag ballistics for big game, but you can also get low-recoil ammo (Remington and Federal) that works fine on deer to 200 yards. You can even get high-velocity saboted .223 rounds in .30-06, for smaller game.
.30-06 is hard to beat as a do-everything hunting round. The military just wanted something that would be smaller so it would be easier to carry more rounds, and so automatic and semiautomatic actions could be more compact. Neither one matters for your purposes, especially since the Remington 7600, the only pump rifle still commonly available, has the same size and weight action for either round.