Popular variations on the .308 case include, in ascending order, the .243 Win, .260, Rem, 7mm-08, and .358 Win. I think the .338 Federal might fall in there too now but I'm not familiar with the cartridge.
I wouldn't go out on a limb and say that these cartridges are better than those built on a '06 case, because a 6.5/06, .280 rem and .35 Whelen are awesome cartridges, but in my opinion with the exception of the .243, they are probably more versatile hunting cartridges than their parent the .308.
I would call the .260 Remington the best of the bunch, with a wide range of projectile choices from 85 to 160gns, which will handle everything from varmints to big game.
The 7mm-08 is a slight step up in bullet diameter and therefore weight, with pills from about 100gn to about 175 being available. A 140gn bullet is fairly standard in 7mm cartridges for deer hunting etc, and they seem to work well. You couldn't really go wrong with the 7-08, which basically duplicates the performance of the classic 7x57 mauser. Tang's suggestion to go with a model 7CDL is also a good one; I have one and they are a joy to carry and shoot.
I nearly bought a 7mm-08 but went for the .260 instead. To me it is the pick of all the .308 based cartridges.