.308;
Pros - Manageable recoil, short action (assuming the rifle is made for the cartridge and not a one size fits all action), ammo is available pretty much everywhere and sensibly priced.
Cons - 1-200ft/s less MV than the '06, less able to cope with the really heavy for caliber bullets (north of 180gn), arguably a bit vanilla, less flat shooting than the '06 in the same bullet weight.
.30.-06;
Pros - 1-200ft/s more MV than the .308, can cope with the really heavy for caliber bullets (north of 180gn), less vanilla, shoots flatter than the .308 in the same bullet weight.
Cons - Bit more recoil, long action, ammo availability not quite as good in some shops.
In the real world there is very little in it. The .308 will chuck a 150gn bullet at roughly 2800ft/s, the .30-06 will chuck it just under 3000ft/s and thats about it. Flat shooting and ability to buck the wind is useful on the hill but in reality its the difference between a 200y zero with 1.5" or 1" elevation at 100y.
Both are great cartridges, both will take anything in the UK. I've used my .308 for driven boar and it did the job just fine.