In most of their senses, there is no difference between skeptic and sceptic. Skeptic is the preferred spelling in American and Canadian English, and sceptic is preferred in the main varieties of English from outside North America. This extends to all derivatives, including sceptical/skeptical and scepticism/skepticism. There is an exception, though: In reference to some 21st-century strains of scientific skepticism, writers and publications from outside North America often use the spellings with the k.
The word comes from the French sceptique,1 which in French is pronounced sep-teek. It has taken several spellings since coming to English in the 16th century, but the modern British spelling was settled by the early 19th