Public variables make sense when the class is nothing more than a bundle of data with no real coherency, or when it's really, really elementary (such as a point class). In general, if there's any variable in a class that you think probably shouldn't be public, that means that the class has some coherence, and variables have a certain relation that should be maintained, so all variables should be private.
Getters and setters make sense when they reflect some sort of coherent idea. In a polygon class, for example, the x and y coordinates of given vertices have a meaning outside the class boundary. It probably makes sense to have getters, and it likely makes sense to have setters. In a bank account class, the balance is probably stored as a private variable, and almost certainly should have a getter. If it has a setter, it needs to have logging built in to preserve auditability.
There are some advantages of getters and setters over public variables. They provide some separation of interface and implementation. Just because a point has a .getX() function doesn't mean there has to be an x, since .getX() and .setX() can be made to work just fine with radial coordinates. Another is that it's possible to maintain class invariants, by doing whatever's necessary to keep the class consistent within the setter. Another is that it's possible to have functionality that triggers on a set, like the logging for the bank account balance.
However, for more abstract classes, the member variables lose individual significance, and only make sense in context. You don't need to know all the internal variables of a C++ stream class, for example. You need to know how to get elements in and out, and how to perform various other actions. If you counted on the exact internal structure, you'd be bogged down in detail that could arbitrarily vary between compilers or versions.
So, I'd say to use private variables almost exclusively, getters and setters where they have a real meaning in object behavior, and not otherwise.
Just because getters and setters are frequently overused doesn't mean they're useless.
