Directory Provider allows a Livecycle Admin to create/sync Users to the Livcycle system, make them part of the system so that they can access the services and perform different activities in the workflow.
On the other hand the Authentication Provider serves as a checkpoint for the said users who are trying to access the system. Authentication Provider asks the user to authenticate itself against various authtication providers(e.g. LDAP, IDP, Db, Custom SPI, Kerberos) before using the Livecycle services.
Please refer the Livecycle Help docs for the detailed description and usage of both Auth Provider and Directory Provider.