The .38 Special case has much more powder capacity than a 9x19 and therefore can hold heavier bullets and drive them to higher velocities. While the heaviest bullet typically loaded in 9x19 is 147 grains, .38 Special cases commonly hold 158-grain bullets and 180-grain bullets. The .38 Special was originally a black-powder cartridge, which explains its large capacity. Early .38 Specials loaded with smokeless powder were limited to low pressures by turn-of-the-century metallurgy. On modern pressure testing equipment, this was equivalent to about 17,000 psi. By 1930 or so, many people were loading .38 Special to much higher pressures. S&W was selling guns for the high-pressure ammo, and there was factory ammo that was probably somewhere around 25,000 psi. Elmer Keith was shooting loads over 40,000 psi. These loads with large masses of bulky slow powders (it was Alliant 2400 in those days) would far exceed the ballistics of modern 9x19mm Parabellum and higher pressure NATO or +P loads. They were more closely comparable to what's now called .38 Super +P or .357 Sig. Not all .38 Special guns could survive with the pressure these loads produced. S&W produced two models based on their large (N) frame from 1930 into the late 60's, and Colt single actions were also capable of holding up to them. With the introduction of the .357 Magnum in 1935, hot loads with the .38 Special case had less attraction. Instead, .38 Special retained favor because it could be shot out of smaller K frame revolvers and it's low pressure and mild recoil made it pleasant to shoot recreationally, and easier to control in police qualifications or combat. Nevertheless, .38 Special gained a reputation for poor effectiveness. This was in large part due to lead round-nosed bullets. Several developments came out of this at different periods. At one point, S&W introduced the massively popular Model 19 -- a (smaller) K-frame .357 Magnum. The Ruger Blackhawk also popularized the .357 Magnum with outdoorsmen in a more affordable revolver. Those guns really increased the popularity of the Magnum but many people still preferred the .38 Special because not only was it easier to carry, it was easier to shoot. Another effort to improve the .38 Special was to increase pressure to 18,500 CUP (today measured as 20,000 psi) and this is given the +P designation. Also, semi-wadcutter bullets and later, jacketed hollowpoints were found to be more effective than round-nosed lead.
Today, the .38 Special +P is limited to 20,000 psi by SAAMI specification. With a 4" barrel, it can propel a 125-grain bullet about 990 fps. This is not quite the velocity of 9x19mm with that bullet weight because 9x19 allows for 35,000 psi. However, .38 Special +P is overwhelmingly popular in guns with little 2" or shorter barrels and that further reduces its velocity substantially (by more than 100 fps with that bullet weight). From these pocket revolvers, it is rare to meet popular standards for penetration and expansion. Some people accept a compromise. They might choose to shoot non-expanding full wadcutters for penetration or accept unreliable expansion and/or penetration.
Factory-loaded .357 can be too stout for comfortable shooting from a pocket revolver. That's why .38 Special +P remains popular. However, you can buy or load your own .357 Magnum that's somewhere between .38 Special +P and the hottest .357's. One way to explain it is like this. The difference between the lightest load in a .357 case and the hottest load in .357 Magnum, is far more than the difference between .380 ACP and .45 ACP. With a .357-rated gun, you can load anywhere in between.
A lot of people that have shot a LCR in 9x19mm have found that 9mm is a pretty stout, high-pressure load when it's not dampened by a reciprocating slide. You can load .357 to have similar or slightly better ballistics than 9x19 -- obtaining an unquestionably effective defensive round with no more recoil than necessary. I recommend a >30 oz. revolver to shoot it. I think this is where the .357 makes the most sense in a defensive revolver: with a load like 158 grain going about 1050 to 1100 fps. This is where we can reliably attain maximum expansion and more than enough penetration. This is distinguishably more than 9mm and with a heavy steel revolver, it shoots very soft compared to a polymer 9. It's still 350 or 400 fps less than the full potential of .357 Magnum from a handgun. Those highest-end loads, maximized by long barrels are more suited to hunting or silhouette competition than defensive use.
.38 Special +P makes some sort of sense from those little pocket revolvers, but the gun has this compromise in performance from the low-pressure cartridge, the short barrel as well as a short sight radius and usually poor sights and heavy trigger. It's a lot of compromises.