How does an ice box differ from a refrigerator?

An icebox is exactly that - an insulated box, nothing more, nothing less, in which a block of ice was placed to cool the air (and contents). In the age before the electrical refrigerator, the icebox filled the same roll.

A refrigerator is similar in appear and construction of the basic "box", but, rather than rely on a cold substance (ice) to cool the interior air, uses an electric pump to decompress one of several gases (formerly freon, now a variety of similarly-performing gases). These gases are decompressed by the refrigerator's main electrical pump, then piped through the main body of the 'frig. Due to the lower gas pressure, they are endothermic, which means, that as the gas expands back up to normal atmospheric pressure, it absorbs heat. The warm gas is then piped back into the electric pump, which decompresses the gas, in an exothermic reaction. Technically, a refrigerator is a simple heat pump; using the decompression/expansion of the gas to draw heat from inside the refrigerator and expelling that heat out into the surrounding air.