I've killed deer with both the .30-'06 (out of several rifles) and a .300 Win Mag. I will have to say that I clicked on the '06 here. Not one animal have I shot with one where the other would have really changed the situation.
That said, let me say this: with a given bullet weight and with a given barrel length, the .300 Win Mag can simply do things that the .30-'06 cannot.
The beauty of the .300 Win Mag is that it is so very flat-shooting. I have a load that I can use to hold dead-on out to 375 yards with, and never rise above nor drop below 5" from P.O.A. At about 200 yards it retains as much velocity and energy as the '06 does at the muzzle. My one and only .300 Win Mag is a 26" Sendero, which I bought specifically for stretching my range over long brush cuts or wheat field shots. Funny thing is, I've never shot any game animal with it as far away as even 150 yds!
Typically, '06 ammo is more easily found, is cheaper, and is found in a wider variety of loads. It recoils a little softer, has a little less blast, and is every bit as accurate --if not more so-- as the .300 Win Mag. All of this translates into a
cartridge that you're more likely to shoot and practice with. There is not an animal on this continent of North America that I wouldn't take with a .30-06 with appropriate loads. (Although I'd for damn sure choose my shots on some creatures like Alaskan brown bear and Alaskan moose and polar bear.) Like so many things with hunting-- it's not the kitchen, it's the cook.
Anecdote:
I took my wife deer hunting the first year we were married. She used a .257 Roberts shooting a lightly-handloaded 100g Nosler Ballistic Tip (~2600 fps) to drop a doe in her own tracks at 90 yards. The next day I shot two large does with my .300 Win Mag loaded with hot 180g Game Kings (~3100 fps). Both were shot at just under 100 yards, and both ran 40 yards into the brush before piling up. I laughed at the time, saying that I had "proven" that the down-loaded .257 Roberts was a superior deer cartidge to the hot-loaded .300 Win Mag!