When an American uses this spelling, it refers to an unwanted consequence of an action, especially one long in the past.
The most common usage is in reference to graphical glitches in computer games or animations. A graphical glitch is referred to as an "Artefact" or "Artefacting", especially with the British rather than American spelling. Most specifically, "Artefacting" references images that are being shown, or pixels getting stuck long after they were supposed to be replaced.
The word can also metaphorically refer to an unwanted consequence that lingers far longer than it should.
Skyrim mudcrabs had weird artefacting in their textures.
*Person A is told to spell artifact*
Person A: A-R-T-E-F-A-C-T
Person B: It's an I, not an E you dumbass.
Person A: No, it's Artefact.
Person B: No, it's actually Artifact. Look at a dictionary and go touch grass you incell.
person 1: i'm british thus i spell 'artefact'
person 2: i'm australian so same
person 3: i'm from new zealand so same person 4: i'm south african so same person 5: i'm canadian so same