There are two types of moderators: Content moderators and Discussion moderators (or "Thread moderators").
Content moderators have power on pages: They can delete/undelete pages, protect/unprotect pages, edit protected pages, and rollback bad edits quickly. I think they can also delete article comments but not edit them.
Discussion moderators have power on discussions of all types - Discussions part, message walls, article comments and blog comments. They can delete/undelete posts, edit posts, see and handle reports (maybe also add discussion categories, I'm not sure).
Admins have the powers of both, and other powers: They can edit the wiki interface (like CSS/JS code or the navigation bar), they can block/unblock users, they can promote/demote discussion moderators, send requests to Fandom staff, and more.
Bureaucrats can promote or demote admins and moderators, and can promote (but not demote) other bureaucrats.
A wiki manager is sort of Fandom staff, so they can do some things that local admins can't. They are there to help large wikis when needed, but are not replacing the admins.