Both spellings mean the same thing: doubtful or unconvinced about a claim. Sceptical is standard in British and Commonwealth English; skeptical is standard in American English. Pick the one that matches your audience and stay consistent.
The core difference
There's no difference in meaning — only in which side of the Atlantic you're writing for.
- sceptical — British/Commonwealth spelling: she remained sceptical about the salesman's promises.
- skeptical — American spelling: I'm skeptical that the plan will actually work.
How to tell them apart
The two spellings trace back to the same word, "sceptic"/"skeptic," plus the ending "-al." American English generally prefers the k, while British and most Commonwealth English keep the c — the same pattern you see across other c/k word pairs in the two varieties.
Where the mix-up shows up most is in mixed-audience writing: a British publication using an American spellchecker, or vice versa, can end up flipping between the two without meaning to. The fix is simple — set your spellchecker's language once and let it hold you to one spelling throughout a document.