I would not hesitate to hunt 99% of the game on earth with a 30-06. While the 300 Win may give you a little extra velocity and slightly flatter trajectory, at hunting ranges on game neither are really relevant. The inch or two in trajectory doesn't matter when your target is between 8 inches and a foot in diameter which the vital area on big game is and a bullet from a 30-06 in the vitals equates to a dead game animal.
Do as you wish but there is a reason the 30-06 is so popular in the world. It works. Period.