I can't open the libraries with explorer or synchronize them which is really inconvenient. Does anyone know why the document libraries on my public site behave differently to those on my private site collections?
What version of SharePoint are you referring to? I assume in your "Public" site publishing feature turned on? Have you checked whether you have similar permission on both libraries.
Do you get any error message when you try to "open with Explorer" - if so can you include a screenshot here?