What's the difference between a galvanic cell and an electrolytic cell?

Really, they are the same thing in construction. The difference is the electrical potentials of the electrodes. In a galvanic cell the combination spontaneously generates a current (a battery).

In an electrolytic cell an "over-voltage" and current is applied to the cells to force the reaction in the opposite direction (recharge cycle of rechargeable batteries, #H_2, O_2# production from water, chlorine production, metal plating).,

There are a few differences, but any similarities as well:

Most importantly, both cells involve two conductors - a cathode and an anode - to move electrons in and out of the cell. In each case, electrons move in the external circuit from anode to cathode. Each electrode is immersed in a conducting solution (or melted ionic compound). Each involves a redox reaction, meaning that an oxidation will occur at the anode of the cell, and a reduction will take place at the cathode.

That's about it for the similarities.

In terms of differences, the most significant is that a galvanic cell utilizes a spontaneous exothermic reaction to provide a steady supply of electrons with potential energy (related to the cell voltage) that will do useful work on some device.

In an electrolytic cell, we are faced with a non-spontaneous endothermic process that we cause to occur by supplying a steady stream of electrons (from a battery, for example). The voltage of the battery must be greater than the absolute value of the (negative) reaction potential that always accompanies a reaction being carried out in an electrolytic cell.

One more difference that frequently causes confusion. The sign convention used for the electrodes. In a galvanic cell, the anode is negative and the cathode positive (such that electrons move from negative to positive). But in an electrolytic cell, these signs are reversed - the cathode is negative and the anode positive.

This is because the cathode of the electrolytic cell is connected to the anode of the battery (and vice versa for the anode of the electrolytic cell, which connects to the battery cathode). This ensures the one thing that must be true. Electrons always leave a cell at the anode and enter at the cathode.

Check out the illustration: