The 22 Mag is the exact same thing as 22 WMR (Winchester Magnum Rimfire).
The 22 W.R.F. (Winchester Rim Fire) is the predecessor to the WMR. It was originally developed for the Winchester Model 1890 Slide-Action Rifle (in the year 1890), and it was loaded with a 45-grain cast lead bullet propelled by 7-grains of FFFg blackpowder. The case dimensions are the same as the WMR except it is .10" shorter. It can safely be fired in any firearm factory chambered for it or the 22 WMR, but the WMR can not be used in a WRF chambered firearm.
Today, CCI manufactures both WRF and WMR ammo, and the WRF can be found with a 45-grain cast bullet, or a 40-grain JSP. Both loads are loaded to comparable pressures and velocities as the original black powder cartridges (so that they are safe to use in any gun capable of using them).
edit: Reading one of the posts below caused a bell to go off in my head... The 22 Remington Special is/was Remington's version of the Winchester 22 W.R.F., and as such, it is completely interchangeable. It too can be safely fired in and 22 WMR chambered firearm.