Ive used and reloaded for both of these for a long time. Here is my 10c not that anyone cares.
They have almost the same powder capacity and are both capable of and suitable for hunting anything anywhere except maybe the big 5 in Africa...and they've both been used many times for all of those even.
8x57 has more recoil and I've never had one that shot anywhere near as well as a 308. Maybe because there are not many great 8mm bullets around...but ive also not had 7x57s shoot as well as the 7-08s ive had so maybe its one of those unexplained things.
Either way...thats my take having hunted, loaded, and shot many a target with both
