The user must be assigned the Site role of Publisher or Interactor. Read this and pay attention to the section "Site roles and permissions": http://onlinehelp.tableau.com/v10.0/server/en-us/help.htm#users_site_roles.htm
This page validates the above:
You have the permission granted at the Project level so that's good. However, the Project content permissions are set to "Managed by the Owner" so that can cause permissioning confusion...
So, if you have the appropriate Site role set then I'm guessing the permission is being denied at the Workbook level of permissions since the partial screen shot is at the Project level of permissions, not the Workbook level of permissions. Go to the specific Workbook and look at its permission rules.
Argh! Yes, I totally forgot to add that bit. Thanks for pointing that out Mark