According to TechNet https://technet.microsoft.com/en-us/library/jj983794(v=exchg.150).aspx the difference is in the intent:
Shared Mailboxes are for external communication (ie help@company.com), Public Folders are for internal collaboration (ie ProjectTeam@company.com)
Shared Mailbox
A shared mailbox is a mailbox that multiple designated users can access to read and send email messages and to share a common calendar. Shared mailboxes can provide a generic email address (such as info@contoso.com or sales@contoso.com) that customers can use to inquire about your company.
Public Folder
Public folders are designed for shared access and provide an easy and effective way to collect, organize, and share information with other people in your workgroup or organization.