You cannot chamber a 243 in a 6mm chamber unless the 6mm chamber is very sloppy.
Reason being the 243 case is larger in diameter at the shoulder than is the 6mm case at that depth.
I have always been a 6mm Remington fan, but I do own both. [3-6mm's, 1-243]
Always felt that the 6mm was more of a reloader's cartridge, whereas the 243 for the factory fodder user.
This is mostly due to the more limited choices in the 6mm permutation.
The practical difference in similar firearms is not a lot [100-150 fps advantage to the 6mm with a 100 grain bullet]
Regards, Dave.