Basically it's like comparing the .30-06 to the .308.
....I've never heard of the 7x57 round.
It is actually more like comparing a 30-06 to a 30-06
The 7X57 was the first successful modern cartridge developed in 1893. When US troops came up against Spanish troops carrying 7X57 rifles they knew we were way behind with our military weapons. Rather than adopting the German design, they simply took the 7X57 case and enlarged it to .30 and stretched it slightly longer to make the 30-03, which was very slightly modified 3 years later to become the 30-06.
Years later the 30-06 was basically shortened to make the 308. The 308 case has been necked up and down to make the 243, 260, 338 Federal and the 7-08.
When the 308 was necked down to 7mm the round had come full circle. After 100 years, the 1st successful smokless powder round had been re-invented. The 7-08 can be loaded in a true short action, but there have been a few 7X57's over the years loaded in some short action guns although most are in long actions. A 30-06 requires a long action.
A 7X57's case length is exactly 1/2 way between a 30-06 and the 7-08, but ballistically the 7X57 and 7-08 are equal. Actually if you stay within book loads the 7-08 is a little faster with equal bullet weights, but not enough to matter. That is why I said it is like comparing a 30-06 to a 30-06.
Picking one over the other is personal preference. Factory loaded 7X57 ammo is probably loaded on the light side, but with handloads it's about a tie. I see more of a selection of facory loads in 7-08 in stores if that is important. A 7-08 wil be in a short action, while most 7X57 guns will be in a long action.
If you handload, and are willing to push the envelope and load rounds above book loads, the greater case capacity of the 7X57 will get you more performance.