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 century. The development of the k spelling is a natural result of English speakers altering the French pronunciation with the first-syllable k sound. The c is silent in many but by no means all English words containing sc, but writers outside North America never got on board with skeptic—that is, until recently, as the sk- usage appears to be growing outside the United States, perhaps with discussion of climate “skeptics” in the media. Moreover, British and Australian skeptical societies—groups that come together to promote science and critical thinking on subjects such as the paranormal—often used the sk- spelling.
For example, these publications from outside the U.S. use sceptic:
A leading climate sceptic patronised by the oil billionaire Koch brothers faced a potential investigation today. [The Guardian]
The days when you could plausibly call yourself a sceptic while refusing to countenance withdrawal from the EU are over. [Telegraph]
But when it comes to The Farmer Wants a Wife, it’s really hard to keep the sceptic fires burning. [Sydney Morning Herald]
And these North American news organizations use skeptic:
A prominent Canadian climate scientist is suing a leading climate skeptic for libel. [New York Times]
Bilingualism skeptic Jim Cougle contends the hearing should be public. [CBC]
The eye, of course, has long been a favorite example for both Darwin proponents and skeptics because of its intricacy. [Forbes]