I think they are certainly different words, however thanks to the wonders of the English language they have over time come to be homophones which in turn became regional preferences — see http://grammarist.com/spelling/artefact-artifact/.
I don't know Latin but I do have google. It would seem that artifact pertains more to the physical, whereby something of an artificial nature is something that is able to be created; artefact pertains more to the abstract, with something of an artefactual nature being something that is able to be *re*created, though it may manifest in a physical form i.e. a malformed product in a factory may be an artifact (object) of an artefact (error) in the manufacturing process. See http://www.worldwidewords.org/qa/qa-art1.htm, where both words are suggested to originate from Latin arte, however the piece claims the direct ancestor of artifice and artificial is Latin artificium, a thing made by skill or art.
To my mind:
Artifacts are typically objects, stemming from intentional or known factors and/or having historical/social value.
Artefacts are typically aberrations, resulting from systemic or unknown factors, valueless or of negative value.
I work in medical imaging, where artefact identification and reduction is very important as some image artefacts mimic pathologies. In other words artefactual anomalies can mimic pathological anomalies.
Other artefacts destroy images. One method of investigating artefacts in medical imaging is to introduce artificial artefacts i.e. artificially adding noise to a digital image to test the efficacy of artefact (noise) reduction tools. An artefact that arises from time to time in medical imaging is motion artefact, resulting in image unsharpness or image blur.
One might also think of things such as cosmic background radiation as being an artifact of the big bang, with the evidence for cosmic background radiation discovered largely from from artefactual noise in radio waves. See: wikipedia Discovery of cosmic microwave background radiation. As such an artefact can become an artifact, where a physical record of this artefactual noise could become a historical artifact.
Perhaps a better distinction between the two words could be achieved through establishing a dichotomy. Perhaps an artifact could be differentiated by being a whole, a result or outcome, discrete, a perceivable thing of a definable origin (it was fashioned or produced) — only valued through context; an artefact could be differentiated by being a part, a residual, in-discrete, an abstract thing of erroneous origin — only perceived through context.