What's the difference between an Analogy and an Allegory?

Both are metaphors. An allegory is an extended metaphor (typically in story form) and represents broad themes or concepts. A simple analogy is a phrase or paragraph that represents something more specific or focused.

The Orwell novel "Animal Farm" is an allegory of Soviet-Stallinsit dictatorship.

Where as a simple analogy might be something like: "Thinking the world would end on Dec 21 because of the Mayan Calendar is like thinking it was in danger of ending last year because your grandmother forgot to change her 2011 calendar of cute puppies to the 2012 calendar of cute kittens until Jan 11th."

Analogy. mathe matical or philosophical comparisons between two things that if two things share truths then it can be compared to a third but related thing that it may be similar.

Allegory, A written or spoken story or parable that uses fictitious characters experiencing fantastical events that are intended to represent lessons learned in real life.