The technology of European handguns seems to be remarkably static from the 16th to 18th centuries, though I think that there must be much more variation than meets the eye. How were pistols in the 1580s different from those made 200 years later?
They were anything but static. From 1580 to 1780, you had tw0 different main types of ignition systems (forgive the wikipedia links, they are sufficient for demonstrating the different lockwork in this case, and giving a broad sense of the development.)
The wheelock was already obsolete by 1580, but was a major improvement over the matchlock, which used a burning slow match (basically a length of nitrate impregnated rope) to ignite the powder. The wheelock used a rotating steel wheel to spark a flint, which in turn ignited the powder.
The next major improvement in ignition was the flintlock which was a much more efficient ignition. In this, a flint is held in a jaw, and dropped against a frizzen (hardened steel plate which sparks when struck) while also moving back the cover over the powder pan, and allowing the gun to be fired.
During the time you mentioned, technology was anything but static, as the wheel lock was pushed out of prominence by various forms of the flintlock, such as the snap lock, snaphance, and the more "modern" Revolutionary War era flintlocks.
Further developments occurred in shape, quality of materials, safety of the locks, and simplification of the lockwork. If you'd like to see some fine 18th century pistols showing a broad variety of shape and design this dealer has a number that make even my cartridge arm loving heart drool.
To sum it up, a person with a pistol in 1580 might have something as obsolete as an old matchlock, or a somewhat obsolete wheellock, or ideally a more modern form of what evolved into the flintlock as we know it. By 1780, this same person would have a wide array of highly refined flintlocks to choose from in a variety of forms, ranging from basic military pistols, to that finest example of the gunsmith's art, the dueling pistol. The basic function of loading the weapon would not have changed, and the concept of a side lock using flint, and a powder primed pan would still remain. The biggest changes came in the refinement of the lockwork, and improvements in internal lock mechanism. By 1780 the technology was very mature, whereas in 1580, the flint ignition technology was in an infancy.
Tl;Dr, pistols changed a lot. The difference between say a pre WWI automobile and a post WWII Packard.
Edit, fixed broken link.