At the tail end of covid, I could not find 9.3 x 62 brass anywhere; I had to make my own out of 30-06 brass. It was easy enough to size up the neck, trim the cartridge to standard 9.3 length, the first shot fire-formed the shoulder out.
Considering the 338-06 uses the same cartridge, I would expect that a bullet of similar weight would perform in a similar manner.
Hodgdon's data bears this out. I looked at 250 gr bullets for both. Velocities were similar. So here is where the difference comes in, the 9.3 x 62 would require about 10 gr more of the same powder, and would do it at about 10,000 psi less.
I think any real advantage for the 9.3 would be with heavier bullets, 250 is the max weight listed by Hodgdon for .338-06 and 300 gr for the 9.3.
I also think the real advantage for the 338-06 would be that lighter bullets can be loaded, and higher velocities (up over 3,000 fps with 160 gr bullets) giving a flatter trajectory; but I think recoil would be a bit on the "sharp" side. Not a big deal when you only need one or two shots.