These centers are differentiated mainly for reporting purposes, but each of them have particularities that will be explained below.
The Customer Center is used to see Transactions, saved searches, and apply payments. Customers can submit cases using the 'Customer Service and Support' feature, and can opt-in/opt-out of campaign emails.
Company contacts can log in to the Customer Center of their Customer.
Only the vendor can log in to the Vendor Center with Vendor Center permissions. From here the vendor can see Transactions, Saved Searches, and Reports.
From the Partner Center, a company can access lead, prospect, and customer records where they are the selected partner. They can also receive commissions, view, edit, and create subpartners and promotion codes, and view all relevant reports.
Advanced Partner Center
The Advanced Partner Center is similar to what an employee would have access to, and is useful for highly active partners who require more access than the Partner Center provides. This center is fully customizable, the user can add or remove access to Records and Transactions via Setup > Users > Roles > Manage Roles. Access can also be granted to subpartners.
Contacts can log in to the Advanced Partner Center of their Partner. This center uses a user license.