These two words mean exactly the same thing – but one is wrong and the other is correct depending on where you are in the world.
Offence with a c is the preferred spelling in Britain and British English. Offense with an s is preferred in America and American English, however.
Sometimes American writers incorrectly use the wrong spelling of the word to distinguish it from another meaning of offense. For example, offence is used to describe a violation of the law while offense refers to when a sports team has a ball. In both cases, only offense is the appropriate spelling in America.
A quick memory trick: Offense with an s is used in the United States.
My name is Rob Bignell. I’m an affordable, professional editor who runs Inventing Reality Editing Service, which meets the manuscript needs of writers both new and published. I also offer a variety of self-publishing services. During the past decade, I’ve helped more than 300 novelists and nonfiction authors obtain their publishing dreams at reasonable prices. I’m also the author of the 7 Minutes a Day… writing guidebooks, four nonfiction hiking guidebook series, and the literary novel Windmill. Several of my short stories in the literary and science fiction genres also have been published.