What is the difference between Standard Controller and Custom Controller extension ?

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 Visualforce 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 Visualforce 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.

https://developer.salesforce.com/docs/atlas.en-us.pages.meta/pages/pages_controller_def.htm