Due to the prevalence of both calibers over the years in police service around the world, there is a treasure trove of real life examples of both rounds used in actual gun fights from which to draw from...
When 38 Specials ruled the roost, the 158 gr LSWCHP +P, known also as the FBI or Treasury Load, was the most effective in actual gun fights. Much more effective than a 380 solid or HP.
Nowadays, with a lot of snubbies being sold, there are other options, like the Speer Gold Dot short barrel loads like their 135 gr load, which should be seriously considered.
I'm sure the 380 has advanced over the years as well, but my preference is for an all steel 38 Special revolver loaded with the FBI load (which is the most proven) or the new Gold Dot load or equivalent.