Energy definitely favors the 22 mag IF you are shooting the standard 40 grain bullets, even from a pistol. Just as important, you also get a jacketed bullet with the 22 mag for significantly better penetration. On the Stinger, you're stuck with the the lighter, soft lead which is not designed for penetration.
There's also a matter of accuracy. In my pistols, the 22 mags have always been quite accurate, but I've never been able to get more than mediocre accuracy with the Stingers, even by comparison to standard 22 LRs.
Of course, it's all a matter of your application. There may not be enough of a difference in some cases to justify stepping up to a 22 mag. (My long way of asking, what you plan to do?)
This is from a standard ballistics chart. Probably from a 20 inch barreled rifle, but okay for comparison:
The Stinger has a 32 grain copper washed hollow point bullet with a muzzle velocity of 1640 feet per second, and muzzle energy of 191 foot pounds.
A (Federal) .22 WMR has a 40 grain jacketed hollow point bullet with a muzzle velocity of 1910 feet per second, and muzzle energy of 325 foot pounds.
With the Magnum you're getting a 25% heavier bullet, going 16% faster, with 70% more energy. Those percentages would be about the same with other barrel lengths. The Stinger will cost about 16 cents a shot, the WMR twice as much.
I agree with NCG, Stingers tend to keyhole a lot, especially out of shorter barrels, and they drop a lot faster than a 22 Mag. About three times as much over 50 yards.