"Archaic" stems from the same root as "archaeology" -- old stuff. An "archaic" word or item would be something out of chronology -- it belongs in the past, but there may not be any replacement for it.
"Obsolete" means replaced. An obsolete entity is something that is old AND has a replacement.
"Automobile" is both archaic and obsolete. It sounds old and was only used during the earlier part of the 20th century, and it has a replacement -- "car."
The CD-player is obsolete -- it has been replaced by iPods and iPhones -- but it's still relatively new; not yet ancient enough to be considered "archaic" and is only called so in a mocking way.
Aztec ruins are archaic -- belonging to the ancient past. They are not obsolete, however, as we still use them to learn about our history.