With modern ammunition, the difference in performance is getting pretty marginal. In theory, the large .45 should create a larger permanent wound channel.
However, look at the three ways to effective stop someone.
Destruction of the central nervous system, aka brain or spine shot. Near instant incapacitation in almost every case. Makes no difference between 9mm or .45.
Major organ failure. Heart, lungs... something that should be working suddenly isn't. A 9mm or .45 in the right spot will do the same thing. The temporary wound channel/expansion for either round is significant.
Fatal blood loss. Probably the only time where the extra size of the .45 would be a clear advantage. However, it really depends on how many holes, and how close to a major artery you hit.
So in general... accuracy and multiple hits is more important than caliber in most of the aspects. I love my .45, but I bought it when .45acp was only a few dollars more a box than 9mm. Now 9mm is between 1/2-2/3rds the price of .45acp on any given day. Cheaper ammo means more range time. More range time is always a good thing.