A common misconception is that .38 S&W was the parent case for .38 Special since both were introduced by S&W. The .38 S&W, while quite popular from the 1870's through the first half of the 20th century, was an evolutionary dead-end as it uses a .361" diameter bullet that, as far as I am aware, wasn't used by any other common cartridge afterward.
.38 Special was actually a lengthened and more heavily loaded version of .38 Long Colt which was the U.S. Army's standard issue handgun cartridge from 1892 to 1909. The standard, at the time, .38 Long Colt loading was 18 gr of black powder with a 150 gr LRN for a velocity just over 700 fps from a 6" barrel while the .38 Special upped the powder charge to 21 gr black powder and a 158 gr bullet for 800-850 fps from the same length barrel (6").
Additionally, .38 Special likely offered substantially better accuracy than .38 Long Colt due to it being designed from the ground up with a bore diameter which matched its bullet diameter. The parent case of .38 Long Colt is, unsurprisingly, .38 Short Colt, but more was changed from the short to long than the length of the case and powder charge. The .38 Short Colt, like many early metallic cartridges, used an outside-lubricated, heel-based bullet not unlike what you'd find in .22 Long rifle ammunition. While its case diameter was the same as it's later offspring, because of its heel-based bullet it actually used a .375" bullet diameter and corresponding bore diameter. When .38 Long Colt was introduced, the benefits of an inside-lubricated bullet were becoming known so rather than change the case diameter, Colt elected to reduce the bullet diameter to .357", but to retain backward compatibility with the old .38 Short Colt ammunition they kept the larger bore diameter and loaded the ammunition with a hollow-based bullet in the hope that it would obturate enough when fired to give acceptable accuracy. This is why many older Colt revolvers have chambers which are bored straight through and lack the reduced diameter chamber throats of more modern guns.
Unfortunately, the hollow-base bullet idea didn't really work as well as Colt had hoped and due to uneven/incomplete obturation, .38 Long Colt revolvers often suffered from mediocre accuracy. S&W's .38 Special revolvers, on the other hand, had proper .357" groove diameters and cylinder throats and thus quickly gained a reputation for excellent accuracy.