I'll take the .38 Special please. Thank you.
Why? I'm glad you asked.
In all the ballistics tests I've seen, .380 JHP ammo typically under-penetrates. It can do well, but seldom gets beyond the 8-10" level. While this may be adequate in many cases it shows one of the cartridge's major faults. It takes a FMJ to penetrate adequately. Even then, the .380 FMJ barely makes the threshold in many cases. At least you need not worry about over-penetration!
The .38 Special out of a 2-inch barrel has rarely been accused of being a barn-burner either. Most of today's +P ammo, when used in a 2-inch revolver, simply pushes the energy level back up to that of a good standard velocity .38 Special round from a 4-inch revolver. Some of it does better than that. But most of the premium .38 JHP loads penetrate well, reaching 12-16 inches, even with a short barrel.
Another advantage of the .38 is the ability to use lead wadcutters for defense. Bullets with a profile like a beer can cut a clean, round hole that promotes bleeding and can allow the introduction of air where it shouldn't be. For a few years in the late 60's and early 70's, it was popular to handload hollowbase wadcutters backwards, with the huge cavity forward. Driving these flying ashtrays at 750-800 fps, they displayed impressive mushrooming in test media. Soft swaged lead wadcutters tended to fragment off the "skirt" area around the cavity though. Hard cast bullets tended to open slower and still expand to impressively.
The .38 in the Bodyguard or Centennial style can be fired through a coat pocket without snagging. Not easy to do with any small pistol.