Public Folders are here to stay and they are fully supported in the new Office and Office 365!
Now that we got that out of our way, I agree there are many scenarios where a site mailbox is a better fit than a public folder. Especially when sharing and collaborating on documents, site mailboxes bring the power of SharePoint (versioning, co-authoring) right into Outlook.
Site mailboxes are scoped to groups of people (teams) while public folders are scoped to the whole organization so there are scenarios for either one and which one to pick really depends on the specific requirements that need to be met.
As for backup/recovery, site mailboxes are treated just like regular mailboxes or shared mailboxes (in fact they are a special type of shared mailboxes) and all procedures that are used for those can also be applied to site mailboxes.
Hope that helps, Alfons