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 mostly disappeared from the language—except in the U.K., where it appears in place of dispatch about a third of the time—and dispatch is the preferred spelling for all senses of the word.
The main exception is in the phrase despatch box, which refers to the lectern in the British House of Commons and the Australian House of Representatives. This use is the main contributor to the relative frequency of despatch in the U.K.
Although dispatch is generally preferred, some British publications occasionally pull out despatch for use 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 (including the same British ones, most of the time) use dispatch for this sense as well as all others—for example:
Campaign surrogates for each of the candidates have been dispatched to the Sunday shows this week. [New York Times]
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]
The report also said Iran was dispatching trucks overland via Iraq to Syria. [Independent]
I intended to effect my interview with the senator with maximum dispatch and return to the pork chop. [Brisbane Dispatch]