It's faster, flatter trajectory and packs a better whallop than the .243.
Reloading data shows very little different in the two, maybe on the edge of 100fps out of 3100 from the muzzle on out (.243 Win 100gr bullet between 2900-3100fps; 6mm Rem 100gr bullet 2900-3250fps). Thats maybe 3% at most, depending on bullet used, and the deer will never know the difference. Since energy qoes as the square of the velocity, the fastest of each shows a ME of (.243 Win) 2133 ft-lbs and (6mm) 2344 ft-lbs for 100-gr bullets. Again, very little difference. The trajectory difference is 0.3" at 100 yards if sighted in at 200 yards, or the same at 250 yards. For a hunter shooting standing, that's insignificant.
.... The round passed through leaving a tiny little hole going in and a quarter sized exit wound coming out.
Not something I'd brag about, because that bullet performed terrible. The idea is to expand and give up ALL of it's energy into the animal to cause the most damage and a quick death. A bullet that passes thru and keeps on going didn't expand well and carried away all the energy still captured by it's remaining velocity. Maybe your shot caused a kill, maybe it was just lucky.
In truth, these two rounds are virtually identical. Except that the .243 is carried in many more small stores/places with more choices in bullet size than the 6mm Rem, which has been tettering on the death bed for years. Don't get me wrong, I'm a fan of any 7x57-based cartridge, as I own two and a .257 Roberts. But it didn't seem like your post was meant to add to the conversation.