In California, the Attorney/Client Privilege is found in California Code of Evidence Section 952.  It provides that all communications between a client and his or her lawyer, that are made in the course of the relationship, are to be kept confidential, if the client so intends.
The Attorney's Duty of Confidentiality is found in the "State Bar Act"  or Business and Professions Code Section 6068.  It provides that all members of the California Bar are to protect the secrets of their clients, even at peril to one's self.
Although these two principles seem to be identical, there are various noteworthy differences between them.  First, the Attorney/Client Privilege protects only information that is learned by the attorney from confidential communications with his or her client or the client's representatives.  However, the Duty of Confidentiality covers information that the attorney obtains from any source -- not necessarily directly from the client.
Also, the Duty of Confidentiality is much broader than the Attorney/Client Privilege.  The former's protections are usually more broadly applied by courts, whereas the latter is usually construed in a manner that would allow the court to learn as much information about the issue before it as possible.
Last, the Attorney/Client Privilege applies to court proceedings where litigation is already taking place.  However, the Duty of Confidentiality is broader, and applies before, during and even after litigation is over.