I've been firing rifles built in both calibers for over thirty years. Much longer than that for the 308. I bought a reamer for the 7mm-08 before we had any rifles in the store. Over the years, I've built quite a few 7mm-08's and found them to shoot quite well but not as well as a similarily constructed 308.
A 7mm-08, built as a heavy varmint class bench rest rifle, can be expected to shoot a bit better than .4 moa. This doesn't mean it won't turn in the occasional sub 1/4 minute group but, on average it just won't quite do it.
A 308, built in the same way, is capable of honest 1/4 moa performance although it is, admittedly, difficult for most people to shoot it to that level.
Of course, the truth is, there is little difference between a 1/4 minute rifle and a 3/8 minute rifle when fired at long range at a 1 minute bull anda cartridge with superior ballistics may even give enough advantage in the wind to make up for the difference. I just think the 308 is easier to tune and is capable of better accuracy than the 7mm-08.
A lot of the reason is due to bore capacity and the suitable powders for various bullet weights. The capacity of the 7mm-08 is just a little bit wrong although some of the newer powders may do the trick. The same sort of problem can arise with the 308 when we are using heavy bullets. Not quite enough capacity for the slower powders and too much for the faster ones. This is one reason I've always kind of thought a 7.62x55 might be just right for pushing 190's for instance. Going in the other direction, hunter class BR shooters have pretty well settled on a smaller case to shoot best with bullets in the 115 to 130 grain range.
Some of the hunter class guys tried the 7mm-08 pretty thoroughly 25 years ago and couldn't get it to work well enough to win even with custom bullets and slower twisted barrels. Some shot really well but couldn't compete with the 308's when it came scoring time.
So, when I say the 7mm-08 won't shoot as well as a 308 I'm not saying it sucks. I'm just saying the 308 can be taken to levels the 7 won't reach.
By the way, I don't think the 6.5's will shoot as well either but they are so much easier to shoot to their potential that they can win over the more difficult to handle 308. Especially when one is shooting from a prone position, reduced recoil is a real help. This is one of the main reasons the 6BR works so well.
Now, when somebody cleans up with a 7mm-08 and kicks my wrinkled butt in the process, I reserve the right to chamge my tune! Regards, Bill.