Reentry permit ( Sai nyukoku kyoka = 再入国許可 )
Assumed that you have status of residence in Japan.
If you have left Japan without any process, your status of residence becomes void.
If you want to reside in Japan again, you newly need to obtain COE and status of residence in Japan.
However such situation is not convenient for you who have some opportunities to depart from Japan.
So, before you depart from Japan, you can visit immigration office and obtain reentry permit for short term.
If you have received this permit and return to Japan within designated duration, you can reside in Japan without application for COE and visa.
Special re-entry permit = deemed re-entry permit ( みなし再入国許可 )
In the case you have status of residence ( excluding short term ) in Japan, if you depart from Japan and return to Japan within one year, you are deemed that you have obtained re-entry permit.
So, you do not have to visit immigration office and obtain re-entry permit.
However at the time you depart from Japan, you need to clarify you have intention to apply for special re-entry permit.
You can not extend this one year while you are outside Japan.
On the other hand, in the case of ordinary re-entry permit, you can extend valid period while you are outside Japan.