They all do use the same diameter bullet, and the "real difference" between them is, each will drive the same weight bullet at a different max velocity. Keep in mind that "generally", the higher the velocity, the more the recoil will be! (assumeing the same weight bullets are used)
For the biggest game (moose for instance) you need to use heavier weight bullets, so the cartridge that drives that heavier bullet fastest, will have the most killing power.
Bottom line is, for deer sized game, they all work fine, so choose the one you like!!
My choise for deer, blk. bear sized animials would be the 280 Rem., as it has less recoil than the magnums, normally comes in a lighter rifle, and has more than enough power!
I shot the Sitka Black Tail in the pict., in the mountains of Kodiak Island (Alaska) with my lightweight 280 Rem.
Drilling Man