A custom controller is an Apex class that implements all of the logic for a page without leveraging a standard controller. Use custom controllers when you want your Visual force page to run entirely in system mode, which does not enforce the permissions and field-level security of the current user.
A controller extension is an Apex class that extends the functionality of a standard or custom controller. Use controller extensions when:You want to leverage the built-in functionality of a standard controller but override one or more actions, such as edit, view, save, or delete.
You want to add new actions.
You want to build a Visual force page that respects user permissions. Although a controller extension class executes in system mode, if a controller extension extends a standard controller, the logic from the standard controller does not execute in system mode. Instead, it executes in user mode, in which permissions, field-level security, and sharing rules of the current user apply.