This is an issue without much real dispute.
Shooting platforms aside, the .38 Special is available with much heavier bullets (up to 180gr) than the .380ACP, which generally means better penetration. This is a good thing for self defense.
Even with the most "specialized" examples of these two calibers' rounds, the .38 Special loads will be faster, for equivalent bullet weights, than the .380s.
Higher velocity & heavier bullet....hmmm....lower velocity & lighter bullet....which is likelier to stop hostilities better?
Some folks believe that velocity makes up for lighter bullets, and they count on frangible bullet expansion to maximal values, but I believe expansion will more likely be suboptimal, so I'll go with the .38 Special.
As for carry platforms, anyone who's actually tried carrying the best compact examples of .380ACP semiautomatic pistols and .38 Special revolvers knows that the latter can be lighter, more comfortable and more concealable.
The revolver obviously has higher overall reliability in short-term use (sudden fast firing of several rounds in emergency conditions) than the semiauto pistol. Reloads are obviously much slower with the revolver, but as a non-LEO, I'm betting on the lack of my own need for such, even though I do always carry at least one reload.
In the end, the only real issue is one of innate capacity; 5-6 rounds for the .38 Special "snubby" vs. 6-9 for the .380ACP pistol.
While it is true that if you ever really need your CCW gun, you'll want as much ammo as possible in the biggest, most powerful caliber that you can control, we all end up making what we consider to be the best-overall compromises we can, else we'd be carrying .50AE Desert Eagles!
As one who has previously carried a .45ACP for decades on a daily basis, I've recently "downgraded" to a "snubby" Ruger SP101 in .357 Magnum, and don't consider myself too shabbily armed. CCW involves compromise, but carrying a "mousegun" in .22LR, .25ACP, .32ACP, and even .380ACP is just more compromise than I'm personally willing for. You may feel differently.