I've hunted and worked with both the 340 Wby and the 338 Rum and both are excellent cartridges that will push the high BC/heavy .338 bullets to good velocities. I'd give the accuracy and velocity edge to the Rum but what advantage the Rum holds is certainly in the "noise".
Like Bob says, the ease of finding 340 Wby brass/ammo is much, much easier. 340 Wby cases can be readily made from 375 H&H brass, though I feel 8 Mag brass or 300 Wby brass is easier to use. One pass through the sizer die makes perfect 340 cases whereas 375 H&H cases need to be incrementally sized down or the case may collapse.
The 338 Edge uses 300 Rum brass and is a little more difficult to find.
Run a 338 Rum or 340 Wby in a 9 lbs. rifle and don't look back.
