ACFT FLYING INTO ACY HAS A LESS SEVERE NMAC WITH ANOTHER ACFT.

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ON INITIAL APCH VECTOR FOR THE ILS RWY 13 AT ACY WE WERE VECTORED ACROSS THE LOC FROM THE SW TO THE NE; (030 DEG HDG ASSIGNED). WE WERE THEN ASSIGNED A 310 DEG HDG AT 3000 FT MSL (L TURN). WE COMPLIED WITH THE CLRNC LIMIT. UPON ROLLOUT IN THE TURN; THE TCASII PICKED UP TFC ABOUT 3-4 MI OPPOSITE DIRECTION AT THE SAME ALT. THE TCASII THEN ISSUED A COMMAND OF MONITOR VERT SPD AND ALMOST IMMEDIATELY THEREAFTER COMMANDED A 'DSND NOW' (APPROX 2500 FPM). WE COMPLIED WITH THE TCASII COMMANDS. ABOUT 3 SECONDS INTO THE RA I NOTICED THAT THE OPPOSITE DIRECTION TFC HAD ALSO STARTED A DSCNT; AT THAT POINT I BEGAN TO MORE AGGRESSIVELY PUSH THE NOSE DOWN (APPROX 3000-4000 FPM DOWN). AT APPROX 2000 FT MSL THE TFC PASSED OVERHEAD; WE THEN LEVELED OFF AND CLBED BACK UP TO OUR ASSIGNED ALT OF 3000 FT MSL AND RPTED THE RA TO ATC. I FEEL THAT A CLBING COMMAND BY OUR TCASII MAY HAVE BEEN A BETTER COURSE OF ACTION; ALSO IF WE HAD A MORE UP TO DATE TCASII IT MAY HAVE GIVEN SOME STEERING COMMANDS. AT NO TIME DID THE ATC FACILITY ISSUE TFC WARNINGS OR ADVISORIES.