In truth, not much. The primary ingredient in both, not surprisingly, is sodium chloride. In fact, the U.S. government requires that any food-grade salt be a minimum of 97.5 percent pure, so any type of salt you consume in the U.S. (and most developed countries) is going to be almost exactly the same as any other type of salt in terms of what it is made up of.
However, “regular” or “table” salt usually contains an anti-clumping agent, like calcium silicate or sodium ferrocyanide, and iodine. Kosher salt usually doesn’t contain either of these things, though occasionally some manufacturers do add an anti-clumping agent. But this is rare, and even when they do, it’s usually much less than the quantity typically added to table salt.
The most striking difference between Kosher salt and regular salt is the grain size. If you look at table salt under a microscope, you’ll see that it has been milled so that the salt crystals look like little cubes. Kosher salt, on the other hand, is larger grained and less processed, and has a more random, crystalline structure.
This size difference is directly responsible for how Kosher salt got its name. Contrary to what is often thought, Kosher salt is not called “Kosher” because the salt itself is specifically certified as such. Rather, Kosher salt got its name because it was used in the process of koshering meat, specifically to remove surface blood from meat. Smaller grained salts, like table salt, have a tendency to dissolve quickly and absorb more into the meat itself. Larger grained salt absorbs the blood and then can be more thoroughly washed off after, removing the surface blood without making the meat overly salty.
Buying any old Kosher salt doesn’t actually guarantee that it is certified kosher (although most mass-produced, food-grade salts, of various varieties, are certified kosher). If you’re curious about whether or not the salt you have is certified kosher, just look on the container for a K or a U that is circled. If it’s there, the salt is kosher, regardless of grain size or added ingredients.