With handloads, the 30-06 will beat the 308 by ~175 fps, but you need to test those loads in the hottest weather you expect to see in the field. For instance, I’m hunting in Texas next month. It will likely be over 100F. We have what passes for a heat wave in Washington this week, so I’m clocking loads to ensure that they’re safe--some powders will take pressures into the stratosphere in summer heat if you're not careful.
I’ve owned and worked with dozens of 308 and 30-06 rifles over the years. I've found two differences that matter:
It's about as easy to find a 0.75 MOA (or better) 308 load as it is to find a 1 MOA 30-06 load.
Some guides will welcome a hunter with a 30-06 but scoff at one with a 308.
I’ve thinned my hunting-rifle herd to a Winchester Model 70 in 30-06 that I got from a friend and a Remington Model 700 in 308 that I bought from a big-box store. Optics are identical. When I got the chance to hunt deer on short notice a few years ago, I took the 308 because I was low on 30-06 ammo and I had enough 308 on hand for the train-up and the hunt itself. These days, I choose the 30-06 because my friend died and I miss him.
That’s as good a reason as I can think of to pick one over the other.