My experience is...
... someone who claims XXX round is a 'sissy' round, simply does not have one and is trying to sell you what he has in stock.
The .308 Winchester is - in military loadings - the equivilent of the military loaded .30-06. Over all, there isn't a lot of difference.
The .30-06 has a larger powder capacity. Everything else being equal, a .30-06 will shoot any given bullet faster than a .308 Win. Perhaps not enough to matter; but a measurable difference. This is particularly meaningful with bullets of greater than 200 grains. The 06 will pitch the heavier bullets distinctly faster.
The .308 Win is more efficient. The powder used (everything else being equal) will give more fps per grain of powder. And, since the .308 Win burns less powder, the recoil impulse (everything else being equal) is less. Probably most important, the .308 Winchester works in a middle length action, where the 06 requires a long action. The .308 Win case has a stouter head and web, too.
Frankly, the two rounds are functionally equal, except for the biggest bullets. Not much to quibble about. Jeff Cooper made the statement the .308 Winchester is the only rifle chambering needed for hunting in the contiguous 48 states, and he's probably right.
But to label it as a 'sissy' round? Well, if he has good selection and good prices, smile sweetly and ignore him gently.