Simple : Fear of retaliation. The Nazis had large quantaties of chemical weapons, but didn´t use them for fear similar retaliatory strikes by the allies, as was the case in WW I. So the lessons from that war were duly noted. The Japanese however had no such fear when subjugating the Chinese, and used gas and blister agents in some instances, and the Italian army used poison gas against the Ethiopians in the Italo-Abyssinian war, but thankfully poison wasn´t used in any other theatres. I can´t view the use of poison gas in concentration camps as chemical warfare, that was just slaughter.

As far as we know, the only country to use chemical weapons in a combat setting during wartime in the European Theatre was, ironically, the United States. The U.S. had developed an experimental mustard and or tear gas impregnated machine gun round. The idea was that the round would penetrate, for instance, the side of a lightly armored halftrack or an Axis bunker and force the defenders out. In the 1944, the new "mustard gas bullet" was tested in the European Theatre. Fortunately, perhaps, for the Allies and the Axis both, it was not a success. The amount of gas was miniscule, and it was found that only huge quantities would actually work. If it did have an effect on the Germans, they didn't consider it a major problem (there were standing orders to retaliate in full if any chemical attacks were launched, and no such retaliatory attacks were planned). As far as we know the US Army didn't use the rounds after 1945. The Japanese, on the other hand, made extensive use of poison gas during the war with China, though intermittently, and generally less commonly after the "official" start of the Second World War in 1939. A number of weapons continued to be developed for use in the deployment and transit of chemical warheads. The Type 94 Tankette, for example, had a special 'chemical trailer' designed for it, with the idea that the tankette would pull the trailer into position, the crew would flip a switch, and enemy troops would be exposed to the gas. We don't think the Japanese ever actually used this in combat. Most Japanese chemical attacks during the Second Sino-Japanese War and (after 1939), the Second World War, were made with artillery shells and WW1 style chemical rounds.

To add on to what everyone else said, Chemical Munitions were not used during WWII because the nature of the war at that time did not lend itself to effective employment of the weapons. The primary purpose of Chemical munitions is to degrade military operations. They achieve this purpose through (1) Contaiminating territory and/or equipment (2) Requireing military units be spread out over a wide target area to prevent a less lucarative target package, which in turn slows down the command and communication process and finally (3) requires troops to take self-protective measrues that inhibits combat operations and functionality of not only solderly operations, but basic human functions like, eating, sleeping, going to the bathroom and the like. You'll notice that I did not say cause casualties. That's because by the end of the first world war, military theorists realized that casualties after an initial attack would be minimal, and as a killing weapon, chemical munitions were in efficient. Additionally, the primary delivery method was via artillery and by 1939 the concept of counter battery fire was developed significantly enough that the long barrages of the first world war, from heavily emplaced guns were obslete. Neither the germans, nor the allies wanted to deal with the issues of chemical weapons in a high mobility environment. Simply put, it would slow everyone down.