"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 The CD-player is obsolete -- it has been replaced by iPods and iPhones -- but it's still relatively new; not yet ancient enough to be considered "archaic" and is only called so in a mocking way.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.

As far as I remember, archaic means something that was common in the past, you can find it in books published long ago. Some men may be able to use it. Obsolete is more strong than archaic, it means that this word (for example) no longer used. Though to be honest, they are almost identical.