The self-driving airplane pretty much exists. Assuming perfect conditions, an airplane can climb to a desired altitude, follow a GPS course, switch to a nonprecision VOR/DME approach, and follow a precision ILS approach (and descent) to within a few hundred feet of the runway. And that's a Cessna 172 with a mildly recent avionics package -- a modern airliner can take you to the runway surface in zero visibility. (Taxiing and takeoff are still manual operations that require more than zero visibility, however.)
The reason a self-flying plane is easier than a self-driving car is because there is much less stuff in the air than on the ground, and everything in the air has a transponder that tells you its exact location and speed. (Not true of small VFR planes, but transponders are becoming more common and if we really wanted to, we could make them mandatory.) Compare this to a car, where there are kids running into the street, traffic lights, other drivers, loads falling off the back of trucks, and so on. The number of things that can suddenly change the driving conditions are dramatic. In the air, that's not the case; the environment is very controlled and you have an extra dimension to navigate in.
With that in mind, we're still a ways away from untrained people getting into an airplane, pressing a button, and arriving at their destination. But honestly, that's details and risk management rather than a fundamental lack of technology. (What happens if the engine fails? What happens if there's a thunderstorm? The corner cases are why you have trained pilots there to assist the computers. If you're willing to die because of a simple mechanical or software failure, then self-flying planes are a reality today. If you're more cautious, then not quite yet :)