Almost NONE if the shot is good. The 2506 is as good as it gets in the deer category, and par-excellent in hunting prairie goats, i.e. antelope. I have owned 8 of them in various makes and models, and have reloaded the cartridge for over 37 years. It is no more prone to ruin meat than a 7mm08, or any other cartridge. A bad shot, is a bad shot regardless of caliber, and the 7mm08, as good as it is, will not keep up with the 2506 when it comes to stretching out for a long shot at a deer, and 7mm08 kills no better than the 2506 in close.
Back to the original question which did not concern the 7mm08. 243 vs 2506!
The 2506 will do anything the 243 will do, and do it better, and at farther distances. The 243 is a fine cartridge, but it is not in the same class as the 2506. The 2506 is INHERENTLY accurate, and is an extreme range gun in the predator category, and a good long range gun in the deer category. I have killed a many of both with a 2506 in the last 37 years, and have owned 5 rifles that were sub-moa shooters, and no, they were not custom rifles. Glass bedded actions, free floated barrels, and a nice trigger, is usually all it takes to make a quality 2506 shoot with good ammo. The 87 grain spitzer's are good whitetail bullets, the 100 grains Nosler partitions are good Muley loads, and a 113 grain Nosler partition is good medicine for elk with good shot placement.