On the rifles I've owned or do own that have octagon barrels, the barrel is thicker than the same rifle in the round barrel version. I believe that octagon barrels:
a) absorb heat better because of their large mass
b) are stiffer (i.e., less prone to 'whipping' when firing); the edges of the flats stiffen the barrel (look at the cross section and note that the edges of the flats stick out further than the center of the flats, almost like external stiffeners).
c) as a result of the above, are more accurate.
The downside is that they are heavier. Of course, you can get the same advantage with a round barrel if you make it heavier, which is what bench shooters do.