What's the difference between the two in terms of how they work?
In both cases, the power steering simply adds some force to the steering when it detects the driver is trying to turn the wheels. Simply, in one case there is an electric motor to add this force, while in the other hydraulic liquid is used.
Is one more efficient at steering than the other?
A hydraulic pump is always in motion while the engine is on. Since any hydraulic system has some internal friction, this means the engine has to consume some power to turn it. The steering pump is permanently connected to the engine crankshaft through a belt and pulley, which also takes some power to keep turning.
On the other hand, an electrical power assist will only draw power when it is actually being used (i.e. the driver is turning the steering). The electrical system also uses an existing source of power, the car alternator - that is already a required piece of equipment for the other needs of the vehicle. But, lacking the extra pulley and pump needed by a hydraulic steering system (and perhaps also an extra belt, depending on engine design), some energy savings are to be expected.
Which one is more reliable (in terms of repairs)? Stats would be nice ;)
No stats ... perhaps because electrical power assist is still a relatively recent technology. Even so, since electric systems have less components and no hydraulic valves and gaskets (so no leaks!), in theory they should be more reliable than hydraulic. In practice, hydraulic power assist has been around for very many years, so it has the advantage of being a well-known, mature technology.
What are the disadvantages and advantages of each one?
The feeling of each system is slightly different from the driver's point of view. From an objective standpoint, perhaps the only statement that can be made is that hydraulic assistance dies instantly when the engine is turned off. Some electrical systems can be programmed to continue running from the battery, though this should be avoided for extended periods of time since the battery will drain. Electric systems can also be perceived to be slightly more reactive to driver input.
From a subjective standpoint, each driver could test different cars with each of these systems to find out which suits best his/her purposes and driving style.