In general, the .38 Special, particularly loaded to +p pressures, is the superior load, speaking purely in ballistic terms. For a point of argument, though, the .380 makes up a lot of ground in being a notably smaller cartridge dimensionally. While the .38 Special provides a ballistic improvement, my Ruger LCP .380 carries significantly easier than my Colt Detective Special .38 Spc. It is lighter, smoother, and has a higher capacity. More modern .38 Spc. revolvers like the Smith & Wesson 637, 638, or 642, or the Ruger LCR, are more apples-to-apples competitors, but even the smallest .38 Specials will not carry quite as easily as the smallest .380 automatic pistols. In addition, it is easier to carry extra ammo for the .380 pistols. Carrying 13 rounds total with my LCP is a breeze compared to carrying 12 with the DS.
However, this is (and this is not fault of yours, but rather of the popularity of the cartridges) a bit too broad a question when you consider the wide variety of loads available for both calibers, particularly for .38 Special. Some of the softer standard pressure loads in .38 Special such as the Federal Hydra-Shok Low Recoil, are relatively light and relatively low-velocity compared to, say, the .380 +p load available from Buffalo Bore. Buffalo Bore's strong .38 Special +p loads, though, knock anything chambered in .380 out of the park.
Speaking very, very generally, .38 Special is about 10-20% more powerful.
I have provided, for your convenience, some comparative ballistics. I have not provided the energy each of these loads produces, since I believe energy is largely incidental in ballistics and sometimes high-energy loads have lower effectiveness compared to lower-energy loads that are more smartly designed. Others take the opposite view - that energy plays a critical role - and they are certainly welcome to post those views. I tend to take a more simple view: the combination of weight and velocity to carry that weight to vital organs is what matters.
From
ATK/Speer @
http://www.speer-ammo.com/ballistics/ammo.aspx:
Speer Gold Dot .380 Auto: Fires a
90-grain projectile at
1040 FPS muzzle velocity
Speer Gold Dot .38 Special (+p): Fires a
125-grain projectile at
945 FPS muzzle velocity
From
ATK/Federal @
http://www.federalpremium.com/products/handgun.aspx:
Federal Hydra-Shok .380 Auto: Fires a
90-grain projectile at
1000 FPS muzzle velocity
Federal Hydra-Shok .38 Special Personal Defense Low Recoil: Fires a
110-grain projectile at
980 FPS velocity
Federal Hydra-Shok .38 Special (+p): Fires a
129-grain projectile at
950 FPS velocity
For what it's worth, in my .380 I carry Remington's Golden Saber, which fires a
102-grain projectile at
940 FPS. Their .38 Special +p load fires a
125-grain projectile at
975 FPS muzzle velocity (from
http://www.remington.com/comparison.aspx)
In my Colt Detective Special, I carry standard pressure rounds due to the older, 1968 D-Frame of the revolver. I carry the Buffalo Bore, which fires a
158-grain Lead Semi-Wadcutter Hollowpoint (LSWC-HP) at
850 FPS muzzle velocity. They have a wide variety of .38 Special and .380 Auto loads in both standard pressure and +p, so visit the website at
https://www.buffalobore.com/index.php?l=product_list&c=1
Both cartridges are considered at the low margin of "acceptable carry calibers". whatever that means. This is very subjective and merely an opinion of mine shared by many. Plenty of people carry .32 ACP, .25 ACP, and even .22LR and feel comfortable doing so.
Both cartridges are also exceedingly popular. so my examples here are a tiny fraction of the loads available.
For excellent reviews of defensive handgun rounds, tnoutdoors9 does superb backyard gel ballistics tests on his YouTube channel
I hope this has been of some help to you. For your convenience, I have provided a random smattering of additional .380 and .38 Special loads for your comparison. I used
www.midwayusa.com for the following ballistics rather than the manufacturer websites
A few (standard pressure) .380 Auto loads:
Hornady Critical Defense: Fires a
90-grain projectile at
1000 FPS muzzle velocity
Winchester PDX1: Fires a
95-grain projectile at
1000 FPS muzzle velocity
Cor-Bon DPX: Fires an
80-grain projectile at
1050 FPS muzzle velocity
A few (+p) .380 Auto loads:
Buffalo Bore Barnes TAC-XP: Fires an
80-grain projectile at
1275 FPS muzzle velocity
Buffalo Bore standard JHP (I think they use Montana Gold bullets these days): Fires a
90-grain projectile at
1200 FPS muzzle velocity
Buffalo Bore standard JHP: Fires a
95-grain projectile at
1125 FPS muzzle velocity
A few (standard pressure) .38 Special loads
Hornady Critical Defense: Fires a
90-grain projectile at
1200 FPS muzzle velocity
Buffalo Bore Wadcutter: Fires a
150-grain projectile at
850 FPS muzzle velocity
Buffalo Bore Barnes TAC-XP: Fires a
110-grain projectile at
1000 FPS muzzle velocity
A few (+p) .38 Special loads
Hornady Critical Defense: Fire a
110-grain projectile at
1090 FPS muzzle velocity
Speer Gold Dot Short Barrel: Fires a
135-grain projectile at
860 FPS muzzle velocity
Winchester PDX1: Fires a
130-grain projectile at
950 FPS muzzle velocity
Doubletap JHP: Fires a
125-grain projectile at
1175 muzzle velocity