I agree with all the answers you have gotten but besides bullet placement the next most important thing is the type of hunting you will be doing.
If you will be hunting from a blind over bait, just about anything from a 243 Win to to a 9.3X74R in the right place will do a fine job.
If the hunt may have long shots then bullet placement may be not such a sure thing and the heavier bullet of the 08 may be a fine choice. However, as some have already said, either one you list will get the job done with a central nervous system or heart lung shot to get you a bear rug.
Someone is going to disagree, but whatever. I did a lot of research between the 7-08 and its father (.308) when I was building my ultralight rifle and the truth to the matter is the 7-08 will do everything the .308 will do with modern bullets. Both cartridges fire light-for-caliber bullets more efficiently than the heavier bullets due to case capacity and when you stick to that philosophy the performance gap dwindles to almost nothing (in practice, not on a balistics table).
Either way, I think you are good to go with either. It sounds like you'd be buying a new rifle to get anything outside of the 7mm-08 and if I were to buy a new rifle and already had a 7-08, it'd have to be something that isnt essentially duplicating what I already have.
pretty new to the bear hunting thing, but not to firearms. I believe either will work just fine with the right bullet in the right spot. I used a 30-06 this spring with a 180gr barnes tsx. He went 15 yards.
I must admit that I have missed a big game animal with a rifle. And not only was it a bull elk, it was the largest bull that I have had in my sights with a tag in my hand. But, as with most "missed shot" stories, I have an excuse. Moisture doesn't really ever bode well for hunting equipment. When moisture (either rain or snow) gets into a firearms barrel, nothing good takes place. A couple rain drops down there can change the point of impact drastically and a barrel jammed with snow after a...