This is just MY opinion ONLY but I would choose the Browning A-bolt without hesitation. There's so many features I like about A-bolts over Remington there's no comparsion. To begin with, the A-bolt has 3 locking lugs, 60 degree bolt lift, extremely smooth bolt operation, detachable concealed magazine, tang safety that lies naturally directly under thumb instead of having to lose grip to manipulate safety on right side, easily adjustable trigger, MUCH better/stronger extractor instead of flimsy stamped as on Remington, from factory glass bedded & free floated, usually but not always better grade of/figure wood, better straighter stock design, safety has 3 separate functions-the first and best locks ALL sear and trigger movement. The other 2 are easily disabled to allow operating bolt while on safe if desired. Need I say any more?
As to the .308 caliber-with the right bullets/placement-it's PLENTY for any deer/black bear walking the earth.