A user who has got the edit access to certain Agile Central projects can't update the Work Items (User Stories/Defects/etc.) however the very same user is able to update tasks which are in the same projects.
Editable detail pages are available in two modes - editing and viewing. Some users don't want the ability to edit a page and instead want to view a work item without having the ability to edit it. You can switch between edit and view mode by clicking the icon next to the name of the work item.
In this case the User had a viewing mode by default switched on which would make EDP pages non-editable while QDP pages (where Tasks were opened) can be still altered (https://help.rallydev.com/new-quick-detail-pages-beta).