I have two .30/06s, but really I could not see any reason anymore to worry if they were .308s instead. I've shot a lot of .308 too.
I really think any differences between them aren't worth worrying about, at least for my own purposes. I accept that the .30/06 is better adapted to heavier bullets, but I only use 150 gn and 185 gn bullets - and not that many of the latter either. I have tried 220 grainers on some big pigs and couldn't see that they offered an advantage over the 150s I usually use. A couple of my mates tried heavier bullets, up to Woodleigh's 240 gn, on buffalo and feral bulls, but they weren't wildly impressed either. I have bigger rifles for big stuff anyway.
For my use, a .308 would make no difference. No real advantage either way really. Sure, a .308 could be made up in a short action, and save 1/2" of length and maybe a tiny bit of weight, assuming we're talking about a repeater, but the difference either way is pretty trivial. My little Browning Stainless .30/06 is light enough, though I was sorely tempted by a mate's Remington 700 Ti that I took out to shoot a bunch of pigs one time. I really don't see any practical difference in killing power or trajectory, and as far as accuracy goes I've shot some exceptionally accurate examples of both. I also don't find the .30/06's recoil an issue, though it is at about this level that I do start to prefer a recoil pad. I can find a greater difference in relative recoil between .308 and .30/06 just by changing propellant, and if recoil was a bother it isn't hard to load or even buy loads with a bit less.
Ultimately if I was to be buying a new rifle it would be a matter of what was available, between these two, rather than insisting on one of these calibres over the other. I just don't think the differences between them are enough to matter.
YMMV
