I'm not certain to which cartridge you refer. There was the old Colt .38 ACP and the later .38 Super. There was a .35 S&W auto cartridge. There's the .380 ACP. There was the .38 AMU.

Is this Smith & Wesson brand ammunition, perhaps in a blue box with white lettering?

I've fired .38 S&W factory loads in .38 Special guns over the years, at least the ones that will chamber. In a given box of factory .38 S&W ammunition I've found that some will chamber and some won't. Some can be forced with varying degrees of success. If one becomes stuck then it must be prodded out with a rod. When firing .38 S&W ammunition in a .38 Special the results are "underwhelming." Accuracy is reasonable at close range and cases eject normally. Nothing detrimental happens and nothing enriching happens either.