I may be oversimplifying things but I do believe that...
An Artefact/Artifact: is some sort of archeologically and/or technologically significant device
while
A Relic: is some sort of archeologically and/or historically significant biological remain.
I disagree, somewhat. An artifact is an object, device or other non-naturally occuring phenomenon but a relic does not need to be biological. It is often is an item with religious significance; pieces of the cross are relics just the same as finger bones of saints. If you want a dictionary definition...
Artifact:
1. any object made by human beings, especially with a view to subsequent use.
2. a handmade object, as a tool, or the remains of one, as a shard of pottery, characteristic of an earlier time or cultural stage, especially such an object found at an archaeological excavation.
3. any mass-produced, usually inexpensive object reflecting contemporary society or popular culture:
artifacts of the pop rock generation.
4. a substance or structure not naturally present in the matter being observed but formed by artificial means, as during preparation of a microscope slide.
5. a spurious observation or result arising from preparatory or investigative procedures.
6. any feature that is not naturally present but is a product of an extrinsic agent, method or the like
Relic:
1. a surviving memorial of something past.
2. an object having interest by reason of its age or its association with the past: a museum of historic relics.
3. a surviving trace of something: a custom that is a relic of paganism.
4. relics.
remaining parts or fragments.
the remains of a deceased person.
5. something kept in remembrance; souvenir; memento.
6. Ecclesiastical. (especially in the Roman Catholic and Greek churches) the body, a part of the body, or some personal memorial of a saint, martyr, or other sacred person, preserved as worthy of veneration.