In modern rifles, there is no difference. They both will drive a 120-gr bullet at close to 3,000 fps, a 140-gr at 2,800 FPS, a 150-gr at 2,650 FPS.
The 7x57mm is short enough to run through a shorter action Mauser or even a Remington Model 7 when using lighter bullets.
The 7mm-08 is a true short action cartridge, based off the .308 Winchester, and is loaded to high pressures, like the .308 is.
In an older 1893, 1894 Mauser, you need to keep the pressures down to 49,000 PSI.
The 7x57R is the rimmed version, for break action guns, and should also be kept to 49,000 PSI.
European standards ( C.I.P.) are for modern rifles and at higher pressures.
USA standards ( SAAMI ) are lower, assuming a lot of older rifles,
7x57mm Max pressure (C.I.P.) 3900 BAR 390.00 MPa (56,565 psi)
7x57mm Max pressure (SAAMI) 3500 BAR 351.63 MPa (51,000 psi) 46,000 CUP
7X57R Max pressure (C.I.P.) 3400 bar = 49312 psi
7x57R Max Pressure (SAAMI ) 3200 bar = 46,000 psi
7mm-08 Max pressure ( SAAMI ) 52,000 CUP.
So the factory loads wring about all the velocity you can expect from a 7mm-08.
The 7x57mm needs handloading to reach full potential with the wide selection of 7mm bullets available.
Another thing to consider is twist rate. Because of so much ammunition being loaded to accommodate older rifles, it is not only the pressure, but they load heavier bullets as the original rifles used, so the twist rate may be 8.66 to 1 or 9:1, for 175-gr and 160-gr bullets. Most 7mm-08 rifles will be 1:10, which works well for 120 through 154 grains. So think about how you will use it, what game, what distance, what bullets you need.