NON-ADHERENCE TO ATC INSTRUCTION CAUSES A POTENTIAL CONFLICT WITH LOSS OF SEPARATION DUE TO HDG TRACK DEV.

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AS WE DEPARTED TEB ARPT ON THE TETERBORO 2 DEP; WE WERE EMPTY OF PAX IN THE CABIN; AS WE HAD JUST DROPPED THEM OFF AT TEB. I WAS FLYING THE ACFT BY HAND (NO AUTOPLT ENGAGED) AND THE PIC WAS TALKING ON THE RADIO. ZNY WAS AS BUSY AS THE NEW YORK APCH CTLR THAT WE TALKED TO DURING THE INSTR APCH TO TEB JUST 20 MINS EARLIER (HE SAID HE WAS WORKING 20 AIRPLANES TO 5 DIFFERENT ARPTS). DURING CLBOUT; WE WERE GIVEN RADAR VECTORS AND ALTS BY THE DEP CTLR WHO ASSIGNED US 8000 FT. AFTER BEING HANDED OFF TO ZNY ON 120.85; WE WERE EVENTUALLY GIVEN A CLB TO FL190 WITH A 'GOOD RATE OF CLB' ASSIGNMENT THROUGH 14000 FT AND 250 KTS MAX SPD. DURING OUR CLB WE WERE GIVEN A TURN TO 260 DEGS (THOUGHT THE PIC) WHICH I DID NOT CLEARLY HEAR OVER THE BUSY SPEAKER. THE PIC BUGGED THE AIRPLANE INTO A L TURN TO 260 DEGS AND POINTED OUT VISUALLY SOME TFC WHICH HE THOUGHT WE WERE BEING VECTORED FOR ON OUR R SIDE. WE ALSO SAW A 737 ON OUR L SIDE; BUT AT FIRST GLANCE HE SEEMED ABOUT 10 MI AWAY AND SEVERAL THOUSAND FT BELOW US. OUR TCASII SYS WAS TRACKING SEVERAL ACFT AT THE TIME ALL AROUND US. WE WERE TOLD TO AMEND ALT TO 14000 FT IN OUR CLB. AS I LEVELED OFF THE ACFT; I WAS LOOKING AT THE ALTIMETER WHEN I HEARD THE TCASII 'TFC -- TFC' TA AND NOTICED A NOW AMBER TARGET ON THE SCREEN AT 11 O'CLOCK AND 5 MI RANGE PLUS 500 FT. I LOOKED OUT AND SAW THE 737 WHICH WAS ALSO MOVING IN OUR WBOUND DIRECTION. THE TCASII GAVE AN RA AND THE TARGET TURNED RED AND RESOLVED US TO DSND. THE PIC PUSHED THE YOKE INTO A DSCNT TO ABOUT 13300 FT AND I NOTICED THE 737 BANK L AS HE TOLD THE CTLR THAT HE ALSO HAD AN RA ON TCASII. AS WE WERE ABLE TO CONVEY OUR ALERT TO THE STILL BUSY CTLR; HE FINALLY GAVE US THE TA OF THE 737 IN OUR AREA. WE TOLD HIM THAT WE HAD RESPONDED TO OUR TCASII RA AND WERE AT 13300 FT ON A 250 DEG HDG. THE CTLR STATED TO 'TURN TO 360 DEGS; I WANTED YOU ON THAT HDG 10 MI AGO.' I TURNED 360 DEGS AND MAINTAINED 14000 FT. AS WE WERE FLYING ON RADIO COCKPIT SPEAKER; I COULD NOT HEAR THE CTLRS AS WELL IN THE BUSY AND NOISY (RUSH OF AIR DOWN LOW AT HIGHER SPDS) COCKPIT. I DID NOT HEAR THE CTLR'S INSTRUCTIONS FOR OUR TURN; BUT SUSPECTED THAT THE PIC WHO WAS WORKING THE RADIOS INTERPRETED THE 360 DEG HDG FOR A 260 DEG HDG WHICH HE ASSURED ME HE READ BACK -- AND WE WERE ON THAT HDG FOR ABOUT 3 MINS.