The .30/06 is my hands down favoirtite cartridge, but I'll try and give you a reasonable comparison in spite of my personal prejudice .
Let's consider the trajectory of the same bullet in both cartridges. A .300 magnum normally drives a 180 gr bullet at 3000 fps, and the .30/06 can make 2800. If we plug those numbers into a ballistics program and choose a flat shooting bullet, say the Nosler Accubond, sighted for maximum point blank range you might not be impressed with the .300 magnum for use at normal hunting ranges.
When both rifles are sighted 4" high at 100 yds, the .300 has an advantage of 1/4" at 300 yards, and 3.5" at 400. I don't like to see my bullet rise that high above the line of sight, but it serves to illustrate the point that at hunting ranges it don't make much difference.
What about the argument that the higher impact velocity will kill better? Again if we choose the same bullet in both rifles, let's say the 180 gr TSX, the bullet is designed so that about one third of it's length will expand and form those deadly petals. In tissue the bullet will open fully with impact velocities as low as 2000 fps. My tests have proven to me that bullets can fail at impact velocities above 2400 fps. When an X bullet "fails", the petals break off but the slug continues to penetrate. Penetration is reduced because there is less mass, and the wound diameter is reduced because the frontal area of the bullet is smaller, so the over all effect is a smaller wound volume and the more time passes before the animal succumbs to blood loss.
I believe that 2400 fps is the optimum impact velocity for a bullet on big game. This velocity ensures maximum bullet upset without failure resulting in optimum penetration and wound volume. Loaded with a 180 gr TSX the otpimum range of the .300 is 300 yards, although the bullet could fail at closer ranges due to the higher velocity. The '06 loaded with the same bullet has an optimal range of 200 yards, and by 300 yards the penetration might come up a bit short of the same bullet from the .300 due to it's lower velocity.
Bullet weight is but one element that determines the recoil we experience when we fire a rifle. Powder charge is another. The difference between the .30/06 and the .300 WSM is not that great, perhaps 5 ft/lbs in rifles of the same weight with maximum charges of Re-19 behind 180 gr bullets. The .300 WSM is a short action round, so the rifle is potentially lighter which would increase the recoil to some degree.
Modern cartridge design is all about accuracy and little else. Most of the newer brand of cartridges have minimum case tapper and sharp shoulders. These rounds will not feed as well as a slender tapered cartridge like the '06 or the .300 H&H. My advice therefore would be not to choose the WSM cartridge if the rifle is going to be called on for a fast repeat shot on big game. On the other hand, if you want a long range cartridge, you could do a lot worse.