I think there is a difference, despite the generally expressed opinion that they are identical words. To my mind, despatch is a verb, as in despatch a parcel or execute someone (sorry fredpuli47 and sandmaster, for different reasons!), and dispatch is an abstract noun similar to the word 'speed', as in to act with dispatch - meaning quickly.
So what if the Oxford English Dictionary disagrees with my opinion! ("There is some difference of opinion on this matter - on the one side there is me, and on the other side there is the rest of the world" to misquote from Blackadder).