Ever try to drive a screw into a board with a hammer? Or drive a big nail into a board with a screwdriver? Both are useful tools, but both have strengths and weaknesses, and you need to select the right one for the application.
Ammo works the same way.
In the Ammo Oracle, we discuss the performance factors that are important for ammo that will be used for human combat (including self defense), with the understanding that the goal is to incapacitate your attacker as quickly as possible, and that your attacker isn't going to stand at attention facing you, which means that you can't count on an ideal, unobstructed frontal shot. These truths factor heavily into the performance criteria we look for.
But, human combat is one of many applications for ammo. Creatures of different sizes and construction will require different performance from a bullet/caliber. A .22LR is plenty of bullet for a squirrel or rabbit, but no where near enough for an elk or bear. On the other hand, a .45/70 is great for elk and bear, but wouldn't be a great choice for turning a rabbit into rabbit stew, 'cause there wouldn't be much left to cook.
Looking at the two loads you are discussing, the .223 load has a lot more case capacity, allowing more propellant to fire a heavier bullet to a much higher velocity. Velocity is important, as the formula for determining the ENERGY of a bullet uses the SQUARE of the velocity.
Kinetic Energy = 1/2 x Mass x Velocity x Velocity
or
KE = .5 x M x V^2
What this really means is that the .223 has *significantly* more energy available to the bullet than a .22Mag at a given distance, plus the bullet is heavier as well. Assuming you have a bullet that is designed and constructed to make good use of that available energy, the .223 is going to be able to crush far more tissue, meaning a bigger wound that will incapacitate more quickly. Actual live-fire testing bears this out.
Remember, too, that a different bullet design (i.e., one that fragments), with that much KE available, can do even more damage than the TBBC bullet in your example. The .22Mag doesn't have enough energy to cause any bullet to fragment.
Does that help?
-Troy