I shoot a .325 WSM in a model 16 savage.
The gains for the 300 WSM are around 250 fps in a 180 grain, and around 350 fps in a 168 grain over the ought six.
One of the cons, is magazine capacity which is now 2 or 3 rounds depending on what make or model you have or will aquire, whereas with the 30-06 it is often 4 or 5 in the mag, as well expect to spend more for ammo, for me I can get a box of .325WSM at shooters edge for about $39 a box, at wholesale it's $75 or something, which is a fair bit more as compared to the 30-06.
And, just because it is a shorter round, doesn't mean it is a lighter rifle, the savages use a larger barrel shank for the WSM's, mine with a 24" barrel isn't any lighter or more nimble then my brothers win mag.