As an owner of both, I only bought the .380 (LCP) because it let me pocket-holster carry it in some situations where I couldn't easily fit one of my 5-shot snubs into a pants/jeans pocket, or at all. The LCP lets me carry a retirement/LEOSA weapon when I don't feel like dressing around a larger gun, and the 5-shot snub isn't quite as easily pocketed.
I prefer the heavier bullet weight loads of the .38 Spl +P, but if the much lighter weight .380 loads are available in situations where I'd probably leave the .38 Spl at home? I'll carry the diminutive .380 and deal with it. (FWIW, it wasn't until I'd borrowed a LCP from a friend, and had been able to see how well a number of them were working, that I even considered going "below" my .38 Spl "minimum" again. The last .380 I'd owned/carried had been more than 25 years earlier.)
Now, trying to compare ballistics between the .380 ACP & .38 Spl is no doubt one of those caliber debates that makes for lots of discussion, magazine sales, etc, especially among internet gun forums. However ...
It really comes down to
how well & effectively the individual
user can use
either that's going to probably be the critical deciding factor. After all, if you can't consistently controllably, accurately and effectively use it under stress, what good is it really going to do?
Both cartridges are reported as seeming to well 'enough' in the area of being used as backup/secondary weapons for LE users, and they're both enjoying another cycle of interest from that market. Both are often considered the 'minimum' caliber among a number of agencies authorizing (or issuing) small guns for use as secondary & off-duty weapons.
The gun companies and ammunition makers have been investing a lot of attention and production effort to filling the demand for both calibers, too.
During one of my recent armorer updates the observation was made by a company armorer instructor that the fastest & largest growing segment of the
commercial handgun market in recent years has been that of the .380 pocket pistol. Go figure. Lots of folks simply aren't interested in getting something that won't slip into a small holster in either a pocket or purse.
In LE hands? It's seemingly neck & neck for both calibers. Both have been involved in reports of having saved the lives of cops, on & off-duty.
Now, I'm not in the business or habit of shilling either specific makes/models of guns or ammunition. I'm an instructor & armorer, not a salesman. I'm more interested in how well someone with whom I'm working on the range can safely handle & operate whatever handgun it is they're using (by restriction or choice), and especially how well they can controllably & accurately use it in whatever qual or training courses-of-fire are being used at the time.
At one time, when we were looking ahead to cover more off-duty/secondary ammo needs for an order, we stopped to take count of what was being carried in the way of .38 & .380 ammo, reviewing qual records. I was surprised (at that time) to see it was pretty much an even split between the calibers. I've noticed more of both since then, too.
So ... Which of those small "pocket gun" calibers that you're willing to lawfully carry concealed, under whatever anticipated conditions are going to be involved, can
you use ... and do so under unexpected circumstances that will probably be chaotic, rapidly evolving, not to mention physically, emotionally & psychologically demanding? Might be in reduced/dim lighting, too, and might involve a situation where you may only be able to use 1 hand.
I don't consider either the .38 Spl +P or the .380, fired from sub-3" barrels, to be exactly "optimal" when it comes to power and potential ballistic performance. I don't try to convince myself that either is what I'd prefer to carry
as a primary weapon if I were to again return to duty and working a plainclothes assignment. That's not why I own and use either of them, though.
For me, for .380 ammo, I've tried a few different major-maker JHP's and some FMJ practice loads. I carry one or another of the JHP's at the moment (either Win RA380T T-Series or Speer GDHP, depending what's available at the moment). ask me next year and the answer may, or may not, be the same ... but I'll still be making sure my skillset is such that I'm able to make effective use of the little .380 pistol, regardless of what's loaded in it.
For my .38 snubs? I have several choices of 125-130-135gr +P loads, a couple brands of 158gr LSWCHP/LHP +P and a 110gr standard pressure load (for an older 37), all of which shoot fine in my 7 snubs. I even have a couple Magnum loads I've found that work well in my pair of M&P 340's (without exhibiting bullet pull), but I commonly carry +P loads in those.
Now, if you want to talk about larger .380's that have 3.5"-4" barrels? I'll carry one of my 3"-3.5" 9mm's.