With the same weight bullet, say 180 grs., the .300 WM gives you about 200-250 fps. more velocity. That means that it's bullet will expand a little further out than the slower, '06 bullet. If that means much to you, then the .300 has an edge.
Since I've had the 160 gr. 7mm Nosler Partition expand quite well at almost 500 yds. when started at 2680 fps out of a .280, I have no doubt that the 180 gr. NP started at 2700-2800 fps. would also.
I would tend to favor the '06 over the .300 due to less recoil in the lighter rifles which I prefer. E