Hoo boy, the wonderful world of cartridge designations. There are several common bullets that share the same diameter, namely .356 inch. These, in approximate order of development .38 S&W, .38 Special, 9mm Parabellum, .357 Magnum, 9mm Kurz/.380 ACP, .38 Super, .357 Maximum, .357 S&WT, and .357 Sig. I may have missed a couple, but these are all the majors. Why .38 and not .356? The .38 S&W was developed from an earlier round, the .38/40, IIRC which did measure .38 inches. Since S&W intended their new cartridge as an improvement, they used the older number instead of a new smaller number, probably as a marketing ploy.
Now for the real differences. The .38 S&W, .38 Special, .357 Magnum, and .357 Maximum are all identical except in one respect, the length of the casing. This prevents loading an overpowered cartridge in a gun that cannot handle it. The differences in the rest, except the .357 Sig, which doesn’t look anything at all like the rest, being what is known as a necked-down case, are also mainly in case length, but there are some minor differences in the design of the base of the cartridge. For the automatic cartridges (which differ from the revolver cartridges I mentioned first in not having a rim), the length progresses from .380/9mm Kurz, 9mm, to .38 Super. I think the S&WT is between the 9mm and .38 Super, but I’ve never actually seen one of these. It was a special target round developed for a line of expensive S&W target autos.
If I didn’t make it clear, the .380 is also known as the 9mm Kurz in Europe. Kurz is German for “short”, which makes since as it is quite a bit shorter than the 9mm Parabellum.
To sum up your question in the simplest possible terms, .38 is usually meant to refer to the .38 Special. .380ACP is a totally different cartridge and the “0” is added to help differentiate it.