The DHC-8, N680MA, was north bound in radar control approximately 8 NM from Anchorage International Airport, and was provided visual separation from the B-737, N673MA, northeast bound on approach to Elmendorf AFB. The B-737 was in clouds at the time of the advisory, and the DHC-8 mistook another B-737 for visual separation. The TCAS alerted the first B-737 pilot of close proximity of traffic approaching at his 2 o'clock position, and he initiated a climb in IMC. A rapid descent was initiated by the DHC-8 pilot after seeing the B-737 emerging from the clouds. These evasive actions occurred approximately 20 seconds prior to the crossing of the tracks of the two airplanes. ATC control of the conflicting airplanes was by the same position controller on separate frequencies. The B-737 pilot was not advised nor aware of the DHC-8 being provided visual separation from his airplane per FAA Order 7110.65G until the TCAS pull-up and visual sighting of the traffic passing under and in front.