Artefact is the most widely used form in Commonwealth countries (excluding Canada); artifact is used in North America. There is no apparent dictionary difference in meaning between the two forms (where I live, artefact is used both for a methodological problem and for an archaeological object).

Given the other answers above, however, it would not surprise me if we are slowly seeing the development of slight variations in meaning between the two spellings on one side of the Atlantic but not the other, much as there is a difference in the UK between enquiry and inquiry, whereas the e- spelling has all but disappeared in the US.

Artefact is the most widely used form in Commonwealth countries (excluding Canada); artifact is used in North America. There is no apparent dictionary difference in meaning between the two forms. Where I live, artefact is used both for a methodological problem and for an archaeological object. Artifact is, however, used in computing, as US spellings predominantly are in many parts of the world.

Given the other answers above, and my own comments, it would not surprise me if we are slowly seeing the development of slight variations in meaning between the two spellings, much as there is a difference in Commonwealth English between enquiry and inquiry, and also as there is between program (computing) and programme (television, theatre).