While the difference is most often cosmetic, an open stud can serve as a connection point for bar elements:
This type of connection is less common than simply using the stud as a stud, but if you are looking to buy elements to build a particular model, you might want to check to see if you need the open stud variety.
In addition to the obvious direct bar connection, open studs can also facilitate connections with the tubes on the underside of 1xn elements. Here's a nice picture showing the bar connection I'm referring to from New Elementary:
This can be used in combination with an open stud to allow elements to connect at a half stud offset. This technique isn't used frequently in sets, but it does sometimes come into play. For example, it is used to fix the top windows in place on the Parisian Restaurant:
The combination of bars on the underside of the 1x6 plates and open studs on the windows allow the windows to be held in place firmly despite the fact that their top stud is offset by half from the regular grid.
jncranton gave lots of great information about connections. Open studs allow for lots of great building techniques. But not all open studs are open simply to facilitate building. Sometimes the studs are molded open for reasons to do with how ABS plastic cools. If the plastic is too thick, it will deform (shrink) as it cools. This is why lots of pieces have open studs, so that they can be formed properly. Not all open studs are open for this reason, but there are lots that are.