The 7mm-08 will definitely outperform the '06 in some areas!
If you like a 7# rifle, and you're shooting within 300 yd at thin skin (deer, sheep and the like) then a 7mm-08 with a 140g will outperform about any combo in 30-06 there is. Lower recoil for similar SD is the key here.
Weight-for-weight you will never beat or even equal a bigger caliber with a smaller case at the same pressure. But for most of NA hunting, a light and handy 7mm-08 can be just the ticket.