I have .257s and 7mm-08.
As stated, if you are shooting the normal bullets for each, such as 115-gr, 129-gr and 140-gr, the recoil will be much lower for the .257 than the 7mm-08.

Of course, it all depends on rifle weight, too.
My big M-700 Classic .257 is very mild, even with hotter loads.
My small 7-lb (with scope) 7mm-08 M-70 carbine has noticeably more recoil with a standard 140-gr bullet, but it's not bad at all.

As I posted here 5 years ago, if you load a 120-gr Nosler Ballistic Tip at 2,650 fps in your 7mm-08, you basically have duplicated a .257 Roberts. You can step it up to 2,800 and still have little recoil, or to the standard 3,000 fps and approach a .25-06. The light load is super accurate, shoots plenty flat to 250 yards, and kills like crazy.