Approach Controller described a conflict event noting the Conflict Alert function does not provide adequate notification of a developing conflict.

Narrative

I was in the process of being relieved from position and was briefing the relieving controller on weather and traffic situations with arrivals and an IFR approach into ZZZ. I was vectoring the B737 for the sequence and told him to descend to 40 and expedite decent through 70 for over flight traffic at 80. I also had departure traffic off southwest bound that I needed to get above the inbound. I started briefing the relieving controller about the weather on final with heavy rain and a -15 KTS reported. I advised him that I was taking aircraft to Runway 32 to avoid the weather; but that we had a situation with an IFR arrival on a GPS approach to a near by airport we had to watch separation with the Runway 32 arrivals. At this point I was distracted; and more time went by than I thought; when I went back to look at the B737 arrival. At this point; no Conflict Alert had gone off; but the aircraft were head on; and 400 FT apart. Both aircraft were on steady courses; and the B737 had a steady descent rate; but I received no aural or visual warning. Why did the Conflict Alert not go off until after separation was lost? Recommendation; a conflict alert probe that works and gives you adequate notification to execute a vector or altitude change. Conflict Alert never went off until after the situation was over! Try to use altitude when you may be distracted; no matter what your experience level is; or how you think an aircraft will perform.