The 7 x 57 is an outstanding cartridge, which is pretty amazing when you consider when it was developed. And the 7 x 57 has a little more case capacity than the 7mm-08. However, the 7mm-08 has a few advantages over the 7mm Mauser. The data for the 7mm-08 was all established for strong, modern actions. In a strong action a reloader can compensate for some of that conservative data when reloading the 7 x 57.
The one area where the 7 x 57 cannot compete with the 7mm-08 is the action length required. The 7 x 57 will not fit in a short action but the 7mm-08 will.
The 7 x 57 is a medium length cartridge. There are some medium length actions out there (SAKO made a nice one years ago) but more often than not the 7mm Mauser gets paired with a long action.
The 7 x 57 IS an excellent cartridge but it is not perfection to the point that its above improvement. If one wants heavy bullets (170ish grains) and has a strong rifle, the 7 x 57 holds a slight edge over its younger rival. In the 140ish grain arena, the 7mm-08 more than holds its own. In a short action, the 7 x 57 can't even enter the race.
I wish rifle manufacturers would still offer a 7 x 57 chambering but I know of no current production rifles that are offered in 7mm Mauser, except for custom makers. The 7mm Mauser isn't going away, it's too good of a cartridge and a lot of people know it. But I do think the 7mm-08 has taken the torch.
While I understand the desirability of of the .308 length cartridges in automatic weapons for use in warfare, I do not understand the clamor for "short action" bolt action rifles. We hunt with bolt action rifles, and the tiny difference in action weight would be easily negated by the hunter spending a few minutes a day on a stair climber. Speed of reloading? If you need a speedy reload after shooting a game animal you have already failed, and the millisecond or two you save by bolting the fractionally shorter bolt through is unlikely to make up for that failure.
The 7mm-08 is a fine cartridge for whatever game we have here in North America before we get to elk. In the hands of a careful and patient hunter it certainly can take elk with the 140 grain bullet. Where the 7x57 shines is in its earlier iterations. Long, heavy for caliber bullets that penetrate and break bones like augers and typically exit. My experience with the use on heavier than deer size animals is thin, but I believe valid having killed 3 elk, 1 moose, and a 1,200 lb. feral Angus bull. All dropped with a single shot to the heart/lung area from quartering away, to quartering toward, to broadside. I shot the bull on the ground a second time, because you know the saying about messing with the bull. My bullet of choice for heavier game is the 175 grain Nosler Partition Semi Spitzer which was discontinued in the 1990's. Not sexy enough I am sure. 48.0 grains of IMR-4350 chronos at 2,640 fps.
All of the extra powder, muzzle blast, recoil, and a 24" barrel to get 200 fps more out of a 7 Remington Magnum never quite added up for me.
Joshua's choice of a 7mm-08 makes a lot of sense over a 7x57 because of the dearth of new 7x57's. I went the custom route back when I chose my rifle. 1939 Mauser action that had already been drilled, tapped, bolt beautifully changed for scope use. I put a Douglas XX premium 22" feather weight barrel in it and had the metal electroless nickel plated. It went into a Rimrock stock with a blind magazine and Model 700 trigger guard. Later pillar bedded by Borden after he bought out Rimrock. With Leupold Vari-X III 1.5x5 scope and a synthetic sling and 4 rounds it weighs at 6 3/4 lbs. As a tool the lines are beautiful, otherwise it fails aesthetically as my taste in custom rifles was corrupted by Jack O'Connor and gorgeous blue and walnut sporters are impractical in the field, and I never got one.
When the neighbor asked me for the use of a rifle to kill his Long Horn/Highland cross steer and Yak that were both too skittish to let the butcher get within 50 yards, I let him use the 7x57 and sure enough he got blood all over the rifle. A wet sponge cleaned it up in mere moments and a wipe with Ballistol took care of the rest.
When it comes to playing with guns and shooting my annual venison, I can fool around with a lot of guns and do. But when I need or want a dedicated hunting rifle to do a job, that 7x57 gets the nod and always will.
Mild recoil, modest muzzle blast, the ability to make brass from a common casing, excellent modern bullets, make Joshua's choice of a 7m-08 a very practical one, and one I'm certain he will not regret.