I'd go with the .38 snubbie. .380's can be effective, but most of the ammo available isn't really suitable for self-defense, IMHO.
I have both, and my first choice for a pocket gun is the snubbie, loaded with +P 158 gr lead hollow points. When I carry the .380 it's loaded with Remington Express FMJ. You probably won't be able to find .380 ACP Remington Express or Winchester Super-X FMJ in a store, but the commonly available FMJ target ammo isn't reliable in my experience.
The Bersa .380 is a fine pistol but, for concealed carry, why carry a .380 that's larger than some 9mm's? One way to pick a pistol is to decide how big of a pistol you want to carry, and then get a pistol that size in a major caliber (9mm and up).