While it looks like they are competitive choices, they really aren’t.
6.8 is a great medium game, intermediate cartridge that works well out of a AR 15 lower. The total package is light, recoil is easy, followup shots quicker and a shooter will try more often. Easier to carry in the field, handier, and all the fun qualities of a carbine - like a lever gun. Parts and availability are high.
A .308 is it’s own build, more expensive as it requires a complete matching lower. The result is heavier, has much more recoil, follow up shots take more time, and are less likely if the shooter dislikes it. It takes more work to carry in the field, the longer length snags more stuff, and has all the qualities of a main battle rifle - work. Parts and availability are less, but its out there.
Ammo? Same price. No, 6.8 is not on the shelf at the 'Mart, neither is .308 in large quantities here. Things are slowly catching up, but my perception of .308 prices is that the surplus is all gone. I know I tried to shoot up my share. And as for buying ammo, the best prices are on the web in bulk. Grabbing another box from the store is a little short sighted, and for the cost of the time, trip, mileage, and other impulse buying, not that cheap after all. Better to have it ahead.
Caliber choices in a one vs. other work best when they both can be shot from the same platform. In this case, they are too different to be comparable. It’s not a question of caliber as much as carbine vs. battle rifle, which brings in a lot of other stuff.
Both will kill deer. I sold my .308 a long time ago, I’m building a 6.8.