No difference, one and the same.
A lot of cartridges have an official designation that includes the name of the company that introduced the cartridge that often gets shortened with common use. Thus, the 44 Remington Magnum becomes the 44 mag, the 243 Winchester becomes a 243, a 327 Federal Magnum becomes a 327 Mag, the 32 H&R magnum becomes a 32 mag and so on. Most of the time this is harmless, albeit a little sloppy, but there are times when you need to specify the whole name to be safe. For example, a 45 Colt is a completely different cartridge than a 45 auto, also called the 45 ACP and then there is the 44 auto magnum which is not the same cartridge at all as the 44 magnum, oops, 44 Remington magnum. Sometimes no rhyme or reason to the way cartridges are named, but not to worry, hang around the shooting world long enough and you'll get the hang of it.
Good question, by the way. I'm sure a lot of folks have wondered.