There is no difference between dispatch and despatch. The latter is an alternative spelling that was common in the 19th century and earlier, but dispatch has gained undisputed dominance in modern English. Despatch has all but disappeared from the language, and dispatch is the appropriate spelling for all senses of the word.
Examples
Although dispatch is generally preferred, some British publications seem to prefer despatch as the noun meaning the act of sending—for example:
The technical fault that delayed the despatch of a government charter flight to Tripoli last night echoed the Foreign Office’s somewhat sluggish response … [Telegraph]
But the despatch by Maurice Parker, the US ambassador to Swaziland, was more direct. [Guardian]
But most publications use dispatch for this sense as well as all others—for example:
Wisconsin state troopers were dispatched Thursday to urge absentee Senate Democrats to return to the capital … [My Fox Boston]
Israel views the dispatch of two Iranian warships to the Mediterranean with “gravity,” PM Benjamin Netanyahu said. [News.com.au]
Instead, when he needs information from the dispatch office at the police station a few miles away, he flips open the laptop mounted next to his steering wheel … [Anniston Star]
I intended to effect my interview with the senator with maximum dispatch and return to the pork chop. [Brisbane Dispatch]
