I believe skyjumper has made a very good point, one which seems to have been missed completely throughout this discussion. Current flow through a wire in and of itself is not dangerous even if you touch a bare wire. As he pointed out, current will only flow through YOUR body if you are at a different potential (voltage) than the wire. If the wire is the neutral or the equipment ground, it is by definition connected to the ground at the main panel, so the wire should be very close to ground potential, which by the way is taken to be zero BY CONVENTION, not because there is some magic about the ground truly being at zero voltage.
So let's discuss real world dangerous situations that can arise, and how an independent ground wire actually functions.
Case 1, very dangerous: The hot wire becomes disconnected from the device. In the case of AC current, the potential of the hot wire varies between about +170V and -170V 60 times per second. Assuming you are at ground potential, and have a resistance of 1000 ohms (could be higher, could be lower), and you touch the hot wire, you are going to get a shock due to the RMS voltage potential difference of 120V, and in fact you will get I = V/R = approximately 1/8 amp of current through you, which can be quite dangerous. If you are wet, and your resistance is lower, you will get more amperage, and you could die. Bad news. Note that the presence of an independent equipment ground in this case does not help you out a bit, and note that the breaker will not trip either.
Case 2: The hot wire frays a little, and a small hair of wire touches the frame of the appliance. In this case, the frame of the appliance is going to be at an elevated voltage, likely somewhere close to 120V RMS, because the hot wire is touching it. If there is no independent equipment ground, there will likely be only a very small current flow through the frame, because the neutral is typically not connected to the frame of the appliance, and the frame is probably not connected to ground well or perhaps not at all. However, the frame is still at a relatively high voltage, so if you touch the frame, you will get a shock, due to the voltage difference between the frame and you. Current will flow through the frame, through you, and to ground through you, and if the wire hair is large enough, and you are wet, you could get seriously injured or killed. If there is an independent equipment ground, it will provide a low resistance path back to ground, and by Kirchoffs law, there will be very little (not zero, but very little) flow through you, because your resistance is higher than the equipment ground wire.
Note that if the neutral were connected to the frame of the appliance, the neutral would carry the stray current back to ground with lower resistance than you, so again you would sustain only a small amount of current flow through your body. In fact, you might not notice at all that the frame was energized. And if the hair of the wire is small enough, you will not trip the breaker.
So I believe the main reason for the independent equipment ground is to ground the frame of the appliance, since we typically do not use the neutral to ground the appliance frame any longer. However, with dryers we often ground the frame through the neutral, if we have a three wire plug.