I had a 30-06 and got rid of it immediately. I wasn't impressed and I didn't give a rat's arse if I could use a variety of bullets with it. So what. Last I saw, it wasn't burning up any shooting competitions. Never heard of a 30-06 sniper, but the .308 has established itself on the competition line, the tactical forces and out in the deer woods. I love the caliber and am trying not to buy another one as we speak. LOL! It's noted as a very good long distance gun on the competition line for a reason. Back in the day at snipercountry, gawd, probably almost 10 years now, we had some serious debates on the -06 vs the .308, but it was apparent which one was better for sniping. So if you want to snipe a deer, and if you go with the consensus, you'd opt for the .308. I don't think the deer cared either way though. They both provide some serious dirt naps.
Anyway, they're both good rounds for deer honestly. I just personally prefer it over the -06. Either way, you place the bullet where it needs to be, you'll be eaten some fine backstraps!
Good luck,
iSnipe