Both the 32 Long and the 32 S&W Long are the same. In your title, it says 32 S&W and 32 Long, which these would NOT be the same. If the pistol is chambered in 32 S&W "Long", then often times people will call it the 32 Long. The 32 S&W (not Long) is a round that is shorter and has less pressure than the 32 S&W Long. I'd have to look it up to be certain, but I'm pretty sure 32 S&W can be fired in a 32 S&W Long, just like the 38 Special is to the 357 Mag.
Quick search on this: Since the .32 S&W headspaces on the rim and shares the rim dimensions and case and bullet diameters of the longer .32 S&W Long, the .32 H&R Magnum cartridges, and the .327 Federal Magnum, .32 S&W cartridges may be fired in arms chambered for these longer cartridges.