Specter and spectre are the same word with two spellings, not two different meanings. Specter is standard in American English, spectre in British English. If you are writing for a UK reader, use spectre; for a US reader, use specter.
The core difference
The words are identical in meaning; the only difference is which country's spelling convention you are following.
- specter — American spelling, used for a ghostly apparition or a looming threat: The specter of another recession hung over the meeting.
- spectre — British spelling of the same word: The spectre of layoffs hung over the office for months.
How to tell them apart
Both spellings trace back to the Latin spectrum, meaning appearance or apparition, the same root that gives us spectrum and, further back through specere (to look at), words like spectacle and inspect. Neither spelling is more "correct" in an absolute sense; the choice simply follows the American-versus-British pattern seen in pairs like theater/theatre and center/centre.
The mix-up shows up most often in editing and publishing, where a manuscript written for one market gets adapted for another and a stray specter or spectre slips through. Setting your spell-checker's dictionary to the right regional English is the simplest fix.