The biggest problem is finding a gun in 6 Remington......
If you are having one built, the 6 Rem will shoot the 80g Sierra Blitz Bt at 3600 fps with Max loads of IMR 4064 and R#19.
The 243 will shoot the 80g Sierra at 3500 fps with Max loads of IMR 4064.
The 6 Rem will shoot the Sierra 60's at 4000 fps using 49.5-50g of 760 into very tiny groups, while the 243 shoots the 60's at 3800-3850 with max loads of 760.
I Have shot out the barrels of many of each on dog towns. Cases have to be trimmed more on the 243 than the 6 Rem, and the Bonanza neck sizer used as a shoulder bumper on both will lengthen case life considerably.
The 6mm Rem does present some problems for guys that like shooting from the magazine. The 6mm Rem is the cartridge that first got me into having zero freebore reamers made. The 6mm Rem is really a long action cartridge if you want to use the magazine and chase the lands as the leade grows. I shot out many Ruger and Remington Varmint's in 6 Rem and shot them as a single shot.
In factory rifles, I think that it is a waste of money buying Lapua brass unless you are a long range match shooter, the chambers are so sloppy in factory rifles that you can not take advantage of the small advantage that the high quality brass gives you. Good brass does not make up for a sloppy chamber. I shoot exceptional groups in my 700 sporters with Rem, Win brass, and PMC brass. If your rifle likes Lapua brass, more power to ya, but you can't make up for a bullet that gets started crooked as it starts to obturate in the lands. I have one 243 that demands Rem brass and another that likes Winchester brand of brass or PMC. These sporter rifles shoot 1 1/2"-2" at 300 yards.
I have had more than a dozen Match quality 243 Ai's and only two 6mm AI's and the 243 AI's win hands down in being user friendly in load development with all kinds of bullets and powders. The 6mm AI's that I have had were picky and had a shorter range of powder charges that would shoot in the Sweet Spot of accuracy.
The 6mm AI is about 200 fps faster on the lighter bullets when compared to the 243 AI when loaded to the point to where you did not have to full length size after every firing. The gap between the two AI's shorten up as you load the 243 AI to full tilt.
The discussion of which is better will go on forever. I would probably make the instant decision based on which one I could get at the best price, knowing that I was going to have to load the 6mm Rem single shot to get the best accuracy out of the rifle.
Good luck!