I've killed four Virginia bears with my 338-06, like your 8mm-06 or a 35 Whelan it's probably more gun than is really needed. Most of the guys I hunt bears with use either a 30-06 or 308 with 180gr bullets. If you are hunting with hounds, which is very common, expect a close range shot.

Bears with a dressed weight of 300 lbs plus are becoming more common every year so use a good bullet like a partition regardless of the gun you pick.
The bottom line is take the gun you want to use because both will work very well.