Okay, when I refer to the ".45" or "9mm" below, I mean .45 ACP or 9mm Luger. Also, I'm considering both rounds in MODERN conventional loadings with modern bullet design unless I specify otherwise. :)
* Stopping Power
In the numbers game, the .45 ACP is a bigger round with more energy so it would get the nod with regards to lab study. In real world use, the .45 ACP still has an advantage BUT one can debate the practicality of the advantage.
With regards to immediate incappacitation, that usually means a shot to the heart or central nervous system (CNS) i.e. brain, spine, whatever. I would argue that a 9mm and .45 have similar success rates when fired directly into the brain, heart, or spine.
Where a .45 has some advantages are when less than vital areas are hit (lungs, liver, extremities). These wounds aren't going to stop a person immediately even if they are lethal hits. What you're looking for on these is shock, quick loss of blood, or both. The .45 is, more than likely, going to make bigger wounds and has a better chance and making a wound going in and one going out (a very good thing). In a life saving situation, though, you may not care if the guy dies 15 minutes after the gunfight or not. What you want is them out of the fight IMMEDIATELY. So, that being said, while we all can agree the .45 has the better potential to kill, in a gun fight to save your life, I'm not sure how much weight that advantage carries. That being said, I carry a .45 sometimes, and that's all my wife carries.
If the round has more energy, it will have more recoil. That's physics. FELT recoil, on the other hand, is different. Much of that felt recoil depends gun design and loadings.
I find lots of people who say a 9mm kicks "more" than a .45 aren't comparing apples to apples. If you dig a little deeper, you'll most likely find something like "my compact 9mm Glock loaded with Gold Dots kicks more than my Kimber full-sized 1911 firing mil-spec FMJ's". Well, yeah, THAT combonation will. That being said, if you take a small frame Taurus in 9mm and one in .45 ACP, and you put Gold Dots in the 9mm and a hefty (but non +P) loading in the .45, you'll probably note the .45 has more punch.
Now, all that being said, I don't find the .45 ACP to be uncomfortable to fire even in practice quantities. I believe that, in felt recoil, there's just a bigger spread between the mil-spec .45 and a Cor-Bon or Black Hills "defense" loading in comparison to a mil-spec 9mm and a hotter defense load. The 9mm is almost always a bit "snappy" while the .45 can run from mild to "neh, I'll save the rest of the ammo for someone I don't like."
* Accuracy and Reliability
Most of this lies strictly with the gun. Neither round has a design advantage with regards to conventional combat accuracy, especially in a compact gun. If there WAS one, odds are none of us would be a good enough shot to exploit that difference, anyway.
With regards to reliability, TECHNICALLY, the diameter to OAL of the 9mm makes it a more reliable feeder in automatics. The .45 ACP is a stubby round, and it doesn't lend itself as well as the 9mm to autoloading. This, of course, can be overcome in mechanical design, so it's no issue in a modern firearm. Crap, it's not an issue in OLD firearms (think 1911 and Hi-Power). For trivia's sake, the 10mm has the most favorable diameter to OAL ratio for autoloading.
I dare say that a Glock in 9mm is no more or less reliable than a Glock in .45 ACP. You can insert Sig, HK, or whatever brands who have a similar family of designs between the two calibers.
To sum up, the .45 ACP is, technically, a more effective round. In practicality, it's also a better round, but one must understand those advantages. The .45 ACP is not going to immediately kill someone in situations where a 9mm wouldn't (save for some barrier in between the gun and target). Having the guy die 20 minutes later on the scene or 4 hours later at the hospital may OR MAY NOT be a good thing. Who knows? It depends on what the guy did when he/she was wounded.
ALL THAT BEING SAID, I carry .357 Sig, .45 ACP, and 10mm almost exclusively. The MAIN REASON I don't carry a 9mm is because . . . I don't own one. Also, with regards to practical carry, I see no advantage in one. I've yet to find a time when I couldn't carry a PT-145 or a G27 where I COULD HAVE carried a 9mm.
I hope that helps!