BE40 DEP FROM TEB MISUNDERSTOOD ATC ASSIGNED HDG AND CONFLICTED WITH TFC.

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WE TOOK OFF FROM RWY 24 AT TEB; AND FLEW THE TEB 5 DEP. AS PER COMPANY SOP'S; WE WERE ALSO FLYING OUR COMPANY NOISE ABATEMENT PROFILE. PER THE DEP; WE WERE CLBING TO 1500 FT ON HDG 250 DEGS. PER OUR SOP; 1000 FT PRIOR TO THE DEP END OF THE RWY; MY CO-PLT WAS BRIEFED TO PULL THE ENGINE POWER BACK TO 80% N1; WHILE I MAINTAINED A SPEED OF 140 KIAS. MY CO-PLT WAS CAUGHT UP IN RAISING THE GEAR; ETC; AFTER ROTATION AS THE END OF THE RWY NEARED. AS I SAW THE '1000 FT PRIOR' COME UP; I PULLED THE PWR BACK ON MY OWN WHILE MY CO-PLT CAUGHT UP; AND CONTINUED TO CLB TO 1500 FT AND THE EXPECTED TURN ON THE DEP. BEFORE REACHING 1500 FT; WE WERE GIVEN A HEADING AND A FURTHER CLB. WITH THE OTHER ACTIVITY GOING ON IN THE COCKPIT; I MISSED THE INITIAL PART OF THE CTLR'S INSTRUCTIONS. I ONLY HEARD A CLB INSTRUCTION TO 3000 FT. MY CO-PLT READ BACK THE TURN TO 'HDG 080 DEGS AND CONTINUE CLB TO 3000 FT.' HE REPEATED THE INSTRUCTION TO ME AND THEN 'TWISTED IN' THE ALT INTO OUR ALT ALERTER AND TWISTED MY HDG BUG TO 080 DEGS. I BEGAN THE TURN. I ASKED TO HAVE THE FLAPS RAISED AND AFTER TAKEOFF CHKS WERE COMPLETED. (THIS IS THE END OF OUR NOISE ABATEMENT PROC.) AS WE NEARED A HDG OF 030 DEGS THE CTLR TOLD US TO STOP TURN AND TURN BACK L TO HDG OF 290 DEGS. HE THEN TOLD US THAT WE HAD BEEN HEADED ON A CONVERGING COURSE WITH THREE OTHER AIRPLANES. HE ALSO ASKED US WHY WOULD HE TURN US E IF WE WERE HDG W. ABOUT THIS TIME; OUR TCAS ALERTED US TO A TA; AND THEN ALMOST IMMEDIATELY; AN RA. SINCE THE CTLR HAD ISSUED US FURTHER INSTRUCTIONS (DESCENT) WHICH WERE IN LINE WITH THE RA; I CONTINUED UNDER HIS DIRECTION. AFTER ANOTHER VECTOR OR TWO; THE CTLR HANDED US OFF WITHOUT FURTHER COMMENT (OTHER THAN MY OFFERED APOLOGY WASN'T NECESSARY). SUPPLEMENTAL INFO FROM ACN 617100: ONCE CHKED IN; WE WERE ASSIGNED AN INITIAL HDG OF 280 DEGS; WHICH I MYSELF (NON-FLYING PLT) AND MY CAPT HEARD AS A HDG OF 080 DEGS. I READ BACK THE HDG BACK OF 080 DEGS AND THE CAPT BEGAN THE TURN.