I own both and have killed many deer with both, but more with the 7mm08.
Both kill deer effectively, but the 7mm08 is definitely superior.
How much more you need depends on how big your deer are, and which species of bears you're likely to encounter. The .243 has failed me far more often than the 7mm08. With a well placed shot, the 7mm08 has never failed to do the job. The .243 may fail to penetrate on an extreamly close shot (under 50yds) and fail to expand at ranges over 300yds. Several aquaintainces used .243's for many years but after many such instances abandoned the .243 for such as the .270 or similar.
East of the Mississippi River, the .243 is much more popular. It kicks a bit less than the 7mm and has been around longer.
If elk, moose, or a likely encounter with a Grizzly/Brown bear are possible, the 7mm08 is MUCH better, though if in such habitat I carry something somewhat larger still. (.300RUM, .338's, or .375magnum, .45/70-400gr @ 2,000fps).
However, I wouldn't purposely use either .243 or 7mm08 for the larger bears.
FWIW; the Georgia state record black bear was killed on a WMA in N.Georgia by a deer hunter with a .243 (100gr Remington CorLokt factory load, single shot). Bear weighed in at check station at 733lbs.... Abnormally large for a black bear, but not impossible for a "garbage dump" bear.... plenty of food, warm winters with long growing season...
I also own a .260. It is much more like the 7mm08 than it is the .243. I still rank the 7mm08 as superior to the .260 except as an extreme long range target round.