As it was said, the two are *almost* the same once the bullet leaves the barrel. If you reload, the extra case capacity of the .30-06 is nice to have, especially with heavy bullets.
If you're really recoil sensitive, go with the .308. The .308 manages to get about the same performance with less powder: From Hodgdon's website, the max load of Varget with a 150gr bullet in the .308 is 47gr, in the '06 it's 51gr. With a 7.5 pound rifle the recoil energy is 20.4 ft-lbs in the .308, 21.5ft-lbs in the '06. The velocity is 2937 in the .308 and 2975 in the '06. Basically the same speed, but the '06 has about 10%*** more recoil, on account of having to use more powder to get the same speed. (Note: I used the 7000fps approximation for the powder exit speed.)
I'm about as big a fan of the '06 as they come, so that's what I'd pick. However, they're close enough (and out here ammo for both is stocked in about the same quantity and variety for each) that you should probably choose based on the gun, rather than the cartridge. If you like rifle A better than B, then don't pick B because it's an '06 instead of a .308 or vice versa.
You won't go wrong either way.
Edit: *** Should be 5%. Oops. As everyone else has said, there are minor differences in recoil, cartridge length, max bullet weight, etc. None of them will mean a lick to Bambi.