I have read a number of reasons for why he never allowed their use but there doesn't seem to be a broad consensus on the issue? Some point to his history in WWI with these weapons while other cite his fear of the repercussions of using such weapons (reprisals by the Allies etc.)
What makes me particularly curious is that towards the end of the war Hitler demanded a harsh scorched earth policy so the invaders would be able to claim nothing (the so called Nero Decree). His mindset seemed to be beyond caring at this point and so why again were no Chemical or Biological weapons unleashed?
(I guess this is two questions then - why did he not generally permit their use and why, when he began to see the pending defeat and became maniacally destructive, would he still not allow them to be used?)
I don't know if this can be a top-level comment but I would like to ask a 3^rd question:
Did the III^rd Reich have a non-trivial amount of chemical/biological weapons at their disposal? (In case they changed their mind/wanted to respond to Allie's use of such weapons)
In Britain during the war, everyone in the country was issued a gas mask and had to carry it around at all times, in anticipation of a German gas attack, even civilians. If the Germans used chemical weapons, the impact of the enemy would have been minimal because of how prepared they were and then they in turn would have needed to adapt an equal level of preparedness as the allies would respond in kind.
It would have been a bad move to escalate to chemical weapons when your enemy is more prepared for it than you are. Towards the end of WW2, Germany was running low on oil and production facilities, which was needed to create more gasmasks. Perhaps they didn't want to unleash weapons that would harm their own forces.
Actually, the introduction of chemical weapons into the war almost happened by accident.
In Eisenhower's "Crusade In Europe" he talks about a surprise German air raid on an Italian port that hit a number of Allied cargo ships currently docked there. At least one of the ships was carrying a large amount of chemical weapons, which the Allies held in reserve. These went off when the ship was struck. Luckily, the wind was blowing out to sea that day, or many Allied soldiers could have been gassed.
Eisenhower didn't seem to think the Germans knew the chemical weapons were there, it was more of a lucky shot, but it could have been the (accidental) start of chemical weapons in the war.
EDIT: Actually, upon further research, Eisenhower lied that there were no casualties. According to other sources, a number of Allied troops were gassed, including 83 dead (more sources in the Wikipedia article on the ship: http://en.wikipedia.org/wiki/John_Harvey_(ship))