I have some experience with the .338/300 Win Mag., plus the .338RUM. If there is a cartridge that comes close to that mythical perfection, it is the .338/300.
The big case ctgs have minimal flexibility compared to the .300 WinMag case. I reamed my own .338 sporter barrel and then worked up loads from extrapolation. I was shooting 65 gr of XMR 4350 in both 300Win and .338 Win with 200 & 250 gr Sierra bullets respectively, so taking that as my floor, the .338/300 with 250 gr bullet is like a kitten with 65 gr. I don't really know what the "max" load is in my rifle. I extrapolated data by comparing the .300 & .330 Dakota and .300 & .340 Wby ctgs. The Dakota cases have the same capcity, the Wby cases have the same dimensions. Basically, the .338/300 will duplicate the .340 Weatherby and be just under the RUM.
Benefits:
Cheaper & better brass is available for .300Win
Burns less powder = longer barrel life
Better magazine capacity
More efficient case, not greatly overbore
Using XMR4350 which likely is not the most effective powder, but I have a lot of it, I estimate I'm getting about 2700 fps with a 250 gr bullet.
With a Near Mfg base and a Leupold Mk4 M3 scope this is a longrange performer in a 12 pound rifle. I am using a basic Sendero barrel profile and H-S Precision stock. With a #3 contour sporter barrel mounted, the rifle is under 10 lbs.
One other nice thing about the .338/300 is the ability to use standard reloading dies with minimal alteration. A Redding .300WM Competition type S Neck Die die will work without alteration, aside from necessitating a reaming a Redding bushing or having one custom made. Similar work enables use of Redding's Comp bullet seater, or adjust a .338 Win seater to fit.
The Flexibility of the .338/300 is what makes it the Real Winner. Unless all you are concerned with is Pure Horsepower, (in which case as has been previously mentioned Go With The .338-378 or Lazzeroni if you want beltless), the .338/300 can work over a wider loading range. I have seen great response from 65-76 gr using 250gr bullets. The 160-300gr range of the .338 makes it Superb for almost any application except plinking; for that try cast bullets!
With power ranking right there with the 340 Weatherby, but greater flexibility, the .338-300 is an All-Around Proven Performer. Likely, barrel life is a lot better than the big-burners. Don't forget, you are burning barrel when you are burning Big Charges Of Powder.
If you are into lugging a 30lb rifle out to the canyon's edge and shooting off a tripod than, hopefully you are buying your barrels half a dozen at a time and enjoying those couple hundred rounds you get from each one after "dialing in your load". But, if you're like me, and enjoy hiking down the trail with a backpack and Need a fast handling relatively light rifle that can do it all; give a looksee to the .338/300 Win Mag.