I created a site template which needed to be applied to other subsites. Within that template was a Web Part of "Content Rollup" - "Project Summary". By default the "Add Task" and "Edit List" options displayed. When the template was applied to the other subsites these options did not display, along with views that were created within other Web Parts but is besides my point.
If I were to re-drop a Web Part on the original site, from the template were it originally displayed, it will show up. Now if I re-dropped it on the cloned subsites from the template were it is not showing up there is no change, the "Add Task" and "Edit List" options do not display.
I have verified that the Edit Web Part settings match, also verified that "Manage Site Features" properties match (e.g. Project Functionality, Access App, Following Content, Getting Started, SharePoint Server Enterprise Site features, Site Feed, Team Collaboration Lists, Wiki Page Home Page, Workflow Task Content Type).