FORMAT recreates the header from scratch, effectively completely obliterating the backup file, or tape. If the backup file or tape is corrupted, this option ignores the corruption, and enables you to reuse the file or tape. From the Microsoft Docs:
FORMAT specifies that a new media set be created. FORMAT causes the backup operation to write a new media header on all media volumes used for the backup operation. The existing contents of the volume become invalid, because any existing media header and backup sets are overwritten.
INIT rewrites field values in the header info. If the device is corrupt, this may not correct the issue. Also from the Microsoft Docs:
Specifies that all backup sets should be overwritten, but preserves the media header. If INIT is specified, any existing backup set on that device is overwritten, if conditions permit. By default, BACKUP checks for the following conditions and does not overwrite the backup media if either condition exists:
- Any backup set has not yet expired. For more information, see the EXPIREDATE and RETAINDAYS options.
- The backup set name given in the BACKUP statement, if provided, does not match the name on the backup media. For more information, see the NAME option, earlier in this section.