If your public folder failed to restore this is probably because the Global admin is not an owner of the public folder and sub-folders.

For public folders to be backed up and restored, it is critical that the Global Admin has Owner level permissions on all folders.

For existing public folders, make sure the Global admin has Owner level permissions at the root level and that these permissions are set on all sub-folders down the entire hierarchy.

When creating a new public folder, set Owner permissions at the root level and all new sub-folders will automatically have the proper permissions.

This is crucial because if a public folder is deleted and the Global admin is not an owner, then it will not be possible to restore the public folder. 

It is also important to note that if a public folder mailbox is deleted, public folders – regardless whether the Global admin is an owner or not – cannot be restored as there is nowhere to restore them to.