Fact is there is a difference between the 3. To compare them is kind of pointless, with a couple of exceptions. At 400 yards the 7mm Rem mag has a pretty good advantage over the 7-08, less over the .280. Assuming you load them all to the teeth, the differences are reasonably incremental from the 7-08, the .280 and the 7 Mag. Fact is, 7mm haters aside, they all benefit from the very high SD and BC of all 7's and the truth is that a real high BC 154 gr out of a 7-08 can do some amazing things given that little case it left. Same for the other 2, but the longer distances give them the slight edge. A good 7mm Rem Mag launching a 160 sturdy BT bullet @ around the 3000 fps mark retains over a ton of energy at well past the 400 yard mark. Combine that with the great penetration and high retained velocity makes it very formidable on anything. Nobody claims that the 7-08 is the cure for magnum-itis, or that it's better than a lot of other choices, but it's mild recoil and typically light weight rifle make it an exceptional choice for most things. An elk gun? There are better, no doubt. Depends where you hunt and how good you are with your rifle. BIG bulls are tough and sure you can kill them with anything given a broadside at the boiler room, but that ain't always the case. For elk like the bigger ones are they deserve more gun at longer ranges and a .300 Mag is up there. A .338 is more like it.
I really like my 7-08 and pack it everywhere but when it's elk or moose killing time, the .300 Mag comes out. My 7mm Tikka is just too damned pretty to keep hunting with, a 1990 M658 Premier. It has to it's credit, however, 3 young bull moose at over 400 yards, one shot kills with primo 160's and I can't argue with that. They'd maybe have been just as dead if I'd used the 7-08 but all 3 were dramatic hits, instantly on the floor that the 7-08 I doubt would have done. The 7 Rem mag is a great and spectacular killer no doubt, but again all 3 rate up there as terrific calibers. I guess the real telling difference is to ask yourself this - if you're in the thick stuff after a moose and a real pissed off big bear decides you're in his space, which caliber would you want of the 3? No contest, IMHO.