This is a hotly debated subject, but the short answer is the cartridge which delivers the greatest energy downrange will be the most effective. The more energy, the greater the damage to the target.
That said, there are many other factors to consider. First off, there's momentum, which you'll never find published but which you can calculate. Momentum is mass times velocity, while energy is 1/2 mass times the SQUARE of velocity. So if you're comparing two cartridges with the same muzzle energy, the heavier, slower bullet will have greater momentum than the lighter, faster one. Conservation of momentum says if two objects collide (say your bullet and a woodchuck), the total momentum of the those two objects after the collision must be the same as before. Consequently, many folks say momentum is "knockdown power", but in practice the things you shoot at are much heavier than your bullet, so momentum does really translate to being able to knock down a human or large animal. On the other hand, momentum is important for mettalic silhouette shooters, who need to knock down their targets without damaging them.
Another thing to consider is bullet sectional density. That's the mass of the bullet divided by it's cross-section area. If you take two bullets with the same energy, the one with the higher sectional density will penetrate further. So a 357 Mag with 158 grain bullets will penetrate further than a 357 Mag with 125 grain bullets. But the lighter bullet, traveling faster, is likely to expand faster and be more effective than the heavier bullet.
Getting back to your .38 Special vs. .22 Mag comparison, you probably shouldn't be comparing with standard-velocity 38 ammo. For anything other than targets, one would use +P .38 Special ammo, which has a muzzle energy comparable to .22 WMR or 9 mm Luger.
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Today's Featured Article - Listening to Your Tractor - by Curtis Von Fange. Years ago there was a TV show about a talking car. Unless you are from another planet, physically or otherwise, I don’t think our internal combustion buddies will talk and tell us their problems. But, on the other hand, there is a secret language that our mechanical companions readily do speak. It is an interesting form of communication that involves all the senses of the listener. In this series we are going to investigate and learn the basic rudimentary skills of understanding this lingo.
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