The question is discussed quite well in Robert Harris' excellent book 'A Higher Form Of Killing - The Secret History of Chemical and Biological Warfare' which was first published in the mid '80s and republished in 2002.
With regards to the Western Allies, while we had biological weapons such as Bacillus Anthracis, it was known that BW were historically less than controllable (which remains the case to this day even with Genetically Engineered bacterial and viral strains); while we had some chemical weapons (Nitrogen and Sulphur Mustards, Phosgene, Cyanides) it was rather a shock to learn about the Nazi nerve agents (Tabun, Soman, Sarin).
Indeed a British chemist (I forget who) had stated just prior to the war that such agents couldn't be produced!
The reason that Hitler didn't use these agents against us is (IMO) that he fervently believed that we did have our own and would retaliate (since there had been a CW 'tit for tat' during WW1). This was because the Germans had become aware of the USA discovering the insecticidal properties of dichlorodiphenyltrichloroethane (DDT) and synthesising it in large quantities. The initial chemical pathways to DDT are the same as those to nerve agents.
As to why he didn't use them against the Russians is, presumably, the belief that, given the nature of the Alliance, the Americans would retaliate.