It depends on a lot.
Assuming either .22 or .177 has enough penetration to get to the vitals and the pellet is placed so as to HIT the vitals, they'll be equal.
If you have a 12 FPE .22, that pellet's only moving at ~600 fps, while a 12 FPE .177 is moving at ~800 FPS. For squirrel and rabbit-sized game, either one will work, but the .177 will be easier to hit the target with, inside 40 yards, because of its flatter trajectory.
Now, if you hit and penetrate the hide of the animal and you're on the margin of clipping something critical, the bigger caliber is better, because it has higher odds of clipping it.
Knock-down power doesn't matter at all. That's not what kills.
Lastly, if you compare an .22 pellet vs. a .177 moving at the same speed, the .22 has a clear advantage, because it will make the larger wound channel AND has the flat trajectory. But then you're talking about a 30 FPE .22 vs. an 18 FPE .177, which is not really fair comparison.
Re. Ted shooting sparrows with an (80 FPE?) .30 cal, I bet the main reason is that no matter what caliber pellet he is shooting, it is moving at between 800 and 950 fps. In that case, the bigger, heavier pellet will be less affected downrange by wind, and will hence be more accurate. (all else considered equal) But the bigger calibers are louder and take more air.
It all depends on what you want to do. Which animals? What range? If you're shooting sparrows and squirrels at up to 15 yards, like I do, anything more than .177 is overkill. Step up to raccoons and possums, and .25 would not be overkill, esp. if you want to have the heart/lung option. I've taken a possum and raccoon at 14 yards with my 18 FPE .177. One shot to damage their brain and take them down, and one shot to finish them. A 30 FPE .22 probably would have not required the second shot. A .177 requires a perfect head shot every time for a humane kill, and they just don't hold their heads still as much as squirrels.