They are different rounds, NOT interchangeable. Among other differences, 8x56 is rimmed, while 8x57 Mauser is rimless. 8x57 is also known as 7.9x57 and 7.92x57.
Sellier & Bellot sells good quality, new production, non-corrosive 8x57. There's also a good amount of military surplus 8x57 available from several countries including Turkey, Romania, and Yugoslavia. This ammo is ok to shoot but it is corrosively primed, so you need to clean the rifle the same day you shoot it, and you need to use something containing water to flush out the corrosive primter residue.
You can use Windex, rubbing alcohol, windshield wiper fluid, USGI Rifle Bore Cleaner, or plain water to flush out corrosive primter residue. Then follow up with Hoppe's No.9 or your favorite powder solvent.