Back in the olden days, Smith and Wesson made a 38, called 38 S&W, and Colt made a different 38 called 38 Colt. The Colt was smaller in diameter. Later the Colt round was lengthened, so they could put more powder in it, to make it more powerful. This new round is the 38 Long Colt. When the Army dropped the Colt 45 Single Action Army, they switched to the Colt double action revolver in 38 Long Colt.
Along came the Spanish American war, and the Moros in the Philippines, who were hopped up on hashish. The Army found that, when they shot them with the 38 Long Colt, it did not stop them and they ran up and carved the soldiers up into little bitty pieces.
The Army re-issued the Single Action Army, because the big 45 stopped the stoned Moros. They went looking for a new gun, which would be chambered in a 45 caliber round. This became the1911 45 automatic.
Smith and Wesson had a brilliant idea. They took the 38 Long Colt and made it longer, so they could put in more powder and make it more powerful. They named their new cartridge the 38 Smith and Wesson Special, and tried to sell it to the Army, but the Army was insisting on a 45, so Smith sold it to police departments all over the country.
Since the original 38 Colt was smaller in diameter than 38 Smith and Wesson (.357" vs. .362"), the 38 Long Colt and the 38 S&W Special are also smaller in diameter.
Because the 38 S&W Special was such a much better round than the 38 S&W, it pretty much replaced it. Very few (if any) guns are still made in 38 S&W. There are only a couple of companies that still load the ammo, and they only load it one way (as opposed to the 38 S&W Special, that is loaded by every ammo maker in the world and in just about every load imaginable).
So much for history. Now to finally answer your question.
What's the difference? 38 S&W is shorter and fatter than 38 S&W Special (just about everyone just calls it 38 Special). They are NOT interchangeable. A 38 S&W will not fit in the hole of a 38 Special, and a 38 Special is too long for the cylinder of a 38 S&W.