They are, in fact NOT the same diameter. Despite what the names may imply. A .500S&W has a bullet diameter of .500" to prevent classification as a destructive device by the BATF in the US.

The .50BMG, is .510" bullet diameter. It is allowed because it's been grandfathered in - it's been around for a very long time. And the few people who can afford to actually own and shoot something in that caliber aren't a whole lot of people - despite what Hollywood, and to be fair some politicians would have you think. Stuff be stupid expensive Yo.

There are no legal or industry required conventions in naming rifle cartridges. You can pretty much call it whatever you want. Even the Europeans do this. You can have a rifle in 8mm Mauser, 8x57 Mauser, or 7.92x57 Mauser. And tehy all refer to the exact same cartridge and are totally interchangeable. There was a brief time with early black powder cartridge arms were about. They would name a cartridge something like .45-70-500 Government. This would tell you the bullet was .45 caliber with a 70grain powder charge, and the bullet weighted 500grains as used by the US government. And then by the late 1800's they somehow lost the thread and went completely off the rails bringing us to today. (I blame the French. Why? No reason, I just irrationally do.) Where many of the "new" calibers are often re-treads with a new name, because marketing.