There's actually a noticeable difference between the two.
.243 has far more support as far as rifles and brass go.
6mm has a hundred or more extra fps in it.
Yep.
The 243 Win is a 308 necked down to 6mm whereas the 6mm Rem and original 244 Rem is a 7x57 necked down to 6mm. Because of this the 6mm Rem has more case capacity than the 243 Win.
When Remington came out with the 6mm Rem it was called the 244 Rem and it only came with a 1-12" twist meaning it couldn't stabilize the heavier bullets for deer-sized game. In the same year, Winchester introduced the 243 with a 1-10" twist. This catapulted the 243's popularity over the 244 Rem. By the time Remington corrected the problem in 1963, it was too late.
Still, from a ballistics standpoint, the 6mm Rem is superior in terms of velocity especially for handloaders. The argument is similar to that of the 7-08 versus the 7x57 with the main difference being no need for considering old weak military actions when working up loads.
I prefer the 6mm Rem if picking between the 2; however, if you're not handloading, factory ammo is much more readily available for the 243 Winchester.
For what its worth, in thumbing through my Hornady 9th Edition book, the 2 are identical in terms of velocity through most if not all of the loads documented. (Keep in mind, Hornady's book rounds off to the nearest 100 fps, which is irritating to me). The Nosler 6th Edition book shows their 55gr BT to yield 4032 fps in the 243 Win's fastest load and 4086 fps in the 6mm Rem's fastest load. On the other end of the spectrum, the 95 gr BT yields 3144 fps for the 243 Win's fastest load and 3261 fps for the 6mm Rem's fastest load.
Some might call this splitting hairs or "marginal" differences. Personally, I think it makes things more interesting, especially in an era when all you see is the same 4-5 cartridges (243 being one of them) with plastic stocks and matte-finished metal.