The 338-06 is a really cool round. Exotic enough that many have never heard of it as Taylorce1 points out, but not hard at all for a reloader. Just run 30-06 brass through 338-06 dies and load. There are a lot of powders that work well and good 338 bullets are widely available.
With lighter 200-225 gr bullets it is right on the heels of a 338 magnum. With heavier bullets it can't keep up, but there aren't many things that it won't kill. I think the biggest advantage is that many rifles, Including the Winchester of Taylorce1's are 5+1 capacity vs only 3+1 for the magnums. If I were in grizzly country I think I'd rather have 2 more rounds than the 50-100 fps offered by the magnum. Lighter recoil is a factor as well. It is going to kick a bit more than typical 30-06 loads, but well below most 338 magnums.
It is not as flat shooting as a 30-06, but pretty close, at least out to ranges most of us are capable of shooting.
Realistically, a 30-06 loaded with 200 gr Accubonds will probably do anything the 338-06 will do with 200 gr accubonds. But many find the 30-06 boring. Doing it with a 338-06 is just a bit more stylish.
I ran across a used custom rifle in the chambering a few years ago. I have absolutely no use for it here in Georgia. I've seriously considered selling it, but just cannot bring myself to do it. I just like the rifle and chambering.