It really depends on what weight bullets you are using.
Recoil is simply conservation of momentum. So mass x velocity of what is leaving the barrel = mass x velocity of rifle, scope etc moving backwards. What some folks miss is that there are two things leaving the barrel - the bullet and the combusted powder. Capacity of the 30-06 and 338-06 is very similar, so combusted powder is little different. The difference tends to be in bullet weight. Traditionally, folks launched bullets in the 210 to 225gr range from the 338-06, so not suprisingly it recoiled a bit. However, with the new copper monolithic bullets (like Barnes), it makes a lot of sense to use lower weights - say 180 to 200gr, because they retain weight like a heavier lead-core bullet of higher sectional density.
If you are launching bullets of similar weights, there is very little difference in the recoil of a 30-06 and 338-06. It is a touch higher, because the 338-06 will launch a 180gr bullet about 100-150 fps faster than the 30-06, so that increase in bullet velocity produces a small increase in recoil by conservation of momentum. However, also consider that on thin skinned game, bore and kinetic energy are really the only significant determinants of killing capacity. And both give you an exponential increase in killing capacity, not a linear relationship. The area of the wound you generate is proportional to the square of your bullet diameter (area of a circle=pi.r^2), so the .338 generates a much bigger hole than the .30-06. And kinetic energy increases exponentially with velocity, but not to mass (KE=1/2mass.velocity^2). Also not that there is very little fall in BC with the 180gr .338 bullet vs the .30 bullet.
So, in short a 338-06 with a 180gr bullet recoils a touch more than 30-06 with a 180gr bullet - say about 10% more. But the there is just no comparison in the terminal ballistics. The 338-06 is just in a completely different class. I will add, it is probably not a class that many folks require in a deer rifle - the 30-06 is clearly already more than adequate. And you can't get more than 100% dead. And if i wanted a shorter barreled rifle, i'd go for the 30-06. I think the .338-06 really deserves at least a 23" barrel.
