The "electrical earth" connection on a wall socket does, eventually, get connected back to a rod in the earth. It also is almost certainly connected to the return on the AC line.
So, I guess the question is, what good might you accomplish if you connected a circuit board ground plane to "electrical earth".
One problem is that most of the circuits in a building also will share that "electrical earth" connection. In a residential house, that would include motors for refrigerators, dishwashers, furnaces, air conditioners, all sorts of other appliances, flourescent lights. And since they are all sharing the "electrical earth" connection, that means that any leakage currents travelling back to earth ground are going to cause ground loops (voltage noise) to be present on the supposed "earth ground".
SO, by connecting the board ground to earth ground, that means that some or all of those building ground loops are going to now be superimposed on your board ground. If it is a stand-alone circuit, then it probably does not care.
However, if it is a circuit that connects to other stuff, then those gound loops can be big enough to cause digital errors. As an example, suppose you have a thermocouple hooked to a copper plumbing pipe in your house, and you run the thermocouple leads back to a controller on a board that is connected to earth ground. The thermocouple signal is in the order of 10's of millivolts, but the copper pipe is probably at zero volts, and the "electrical earth" most likely has 5 volts pk-pk of 60 Hz noise on it. So you are burying your desired signal in a noise that is 500 times bigger! You might get big problems from such a circuit.