For the noun meaning a ghostly apparition or a haunting or disturbing image, American writers use specter. Everywhere else, spectre is the preferred spelling.
Both spellings are several centuries old, but spectre prevailed everywhere until the middle 20th century, when the American-style spelling became prevalent in American English. Today, that spelling is also gaining ground in British English, but spectre is still far more common.
This ngram graphs the use of specter and spectre in a large number of British texts published between 1800 and 2000:
And this ngram graphs the words’ use in American texts:
On the negative side, a writer again tries to scare old people with a phony specter of “death panels.” [The Spokesman Review (U.S.)]
A major drop in the city’s population since the 2000 Census had raised the specter of the city losing one of its six Senate seats. [Washington Post]
But today, a specter is haunting America: A sense that we will no longer be THE dominant world power. [CBS News (U.S.)]