There have been many tests done with very similar results. A 30-06 or 300 magnum when loaded with 200-220 gr Nosler partitions proved to be very effective. They actually outperformed such rounds as 338 win mag, 350 mag, 45-70, 12 ga slugs 338-06 and 35 Whelen. The 308 was not included in any of the tests I've seen, but I wouldn't be surprised it it were equal to the 30-06 and 300 mags when loaded with the same bullets. Like HiBC stated the differnce is range, not close range power. The 300 mags shooting the same bulles 200 fps faster did not perform any better than the 30-06. Dropping down another 100-150 fps may not matter either. These results were only with the heavy Partitions. 180 gr Partitions were included in the tests, they were still acceptable, but no where nearly as good as the heavy bullets. Especially the 220 gr Partition.
The testers had to move all the way up to 375 H&H mag to see any measurable improvement in performance over a 30-06. The Alaska department of wildlife conducted one of the tests and concluded that a 375 mag was the caliber of choice for large bear protection if someone could handle the recoil. A 30-06 was their 2nd choice, and their 1st choice if the recoil from a 375 was too much. The 30-06 also offers the advantage of more rounds available in the guns magazine over magnum chamberings.
I'll include a link to the Alaska test. It is a bit dated, from 1983, but the same tests have been duplicated by several gunwriters more recently using some more modern loadings with almost identical results.
This takes a few minutes to open and is a long document, but a good read.