List Content Type vs Site Content Type

So what is the difference between both?

Update an existing content type that is being used in a list. (Add a new column, edit an existing column or remove one)

Go to the list settings. You will still see the old content type with out the newly added fields or still with the deleted fields. Delete the existing content type (Make sure that there are no list items that use this content type). Add the content type again from the list of existing content types. Also make sure that you remove any references to the columns (such as from views, etc..) and delete the list columns that have been changed. Add the content type again and you will see all the deleted columns again.

So what was the catch?
The content type that was modified was a site content type. SharePoint creates a list content type based on the site content type each time a content type is added to a list.