Endo/apico. Endodontic retreatment usually means re-doing the root canal. An apicoectomy is a surgical procedure to treat the end of the root. This may involve manually cleaning out infected tissue that won't heal. Often times, the tip of the root is removed, and sometimes a filling is placed in the tip of the root to seal the canal.
Similar. Retreatment is just what it says. Unsuccesful root canal fill is removed, rootcanal is redone and then new gutta purcha (endo filling material) is placed in the newly filed canals. In an apicoectomy the root tip is surgically removed and a cement or other filling material is used to plug the freshly cut root tip. Both correct an usuccessful root canal.
Another difference. Retreatment is typically a 'non-surgical' procedure, while an apicoectomy requires an incision to expose the tip of the root so it can be sectioned and removed. Both can be highly successful under the right circumstances. I usually opt for retreatment first, as it is able to clean the canal space. If retreatment is unsuccessful but done well, the apico has a much better chance of success after.