With factory ammunition under 180 gr. there is no real odds,and a .308 would likely be the best bet.
Having said that my personal preference is the 30-06. The reason the .308 will run with the 30-06 is the max.pressure is higher for the .308. This was done due to older rifles kicking around chambered for 30-06.
In modern rifles (like my Mod.70) equivalent pressures are quite safe,I have one standard load for my 06 with a 150 gr. bullet that chronos at slightly over 3000 fps. This is a very accurate load,and I use it for hunting and target work.
I have other rifles for other applications,but this rifle and load are high on my list of favorites.