As hunting cartridges, there is probably not a dimes worth of difference between them.
The 7mag was the first magnum rifle cartridge that I began reloading, back in '76. The 7mag operates at higher pressure and likely will have shorter barrel life if you do high volume shooting.
I grew up shooting 30-06, my father, career military, before his second trip to Nam came home one day with back bumper of our '58 Chevy wagon dragging the ground. He must have had 10,000 rounds of surplus 30-06 in the back. It was loose in GI ammo cans. All mixed, ball, AP and tracer. He soon bought an Interarms MKX, in 30-06, a bargain rifle at the time. I was 12, this was in 1965. I never hunted with that rifle, but my father did and always seemed to bring home meat. I hunted whitetail in South Mississippi once with the 7mag and took a 60 shot on a smallish buck with Speer 140gr Jsp, which did so much meat damage at that range that I took my 30-30 after that.
I think that a reloader might find the 7mag a little more satisfying, but either 7mm or 308 has more bullets available to the reloader than any other caliber that I can think of.
With a decent recoil pad I never found either to be offensive to shoot in that respect. I dont think you can go wrong with either one if you learn to shoot it well.