The attorney-client privilege and the work product doctrine provide entirely separate protections. Communications or documents might be protected by both, one or neither.
For some reason, courts sometimes decline to analyze one of the claims if they find the other applicable. See, e.g., In re Managed Care Litig., 415 F. Supp. 2d 1378, 1381 (S.D. Fla. 2006) (“Because the attorney-client privilege affords greater protection, and because the Court concludes that the attorney-client privilege protects the report from disclosure, the Court need not address the work product doctrine.”); Williams v. Sprint/United Mgmt. Co., No. 03-2200-JWL-DJW, 2006 U.S. Dist. LEXIS 4219 (D. Kan. Feb. 1, 2006) (finding the attorney-client privilege applicable and therefore not considering the work product protection). This might save the court time, but could prejudice the litigant asserting the protections. For instance, the adversary might seek an interlocutory review or ultimate appeal of the privilege ruling ‑‑ leaving the owner of the protections unable to establish the work product protection because there is no trial court record or finding dealing with that separate protection.
Litigant in courts that are inclined to stop the analysis short of addressing both protections should gently remind the judge that they are entitled to rulings on both protections.