FLC DEV FILED BY ATC ON B767 FLC. FO USED THE WRONG SID FOR FMC AND VOR RADIAL INPUT. CAPT FAILED TO CATCH ERROR ON PREFLT. POTENTIAL CONFLICT OCCURRED.

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DURING CLBOUT AT JFK A CONFLICT OCCURRED WITH ANOTHER ACFT. WE WERE CLRED TO FLY THE KENNEDY 7 DEP AND THE BREEZY POINT CLB. DURING PREFLT; I CHKED THE RTE IN THE FMC AGAINST THE FLT PLAN AND EVERYTHING MATCHED. THE FLT PLAN STARTED AT ROBBINSVILLE SO I STEPPED THROUGH IT WITH THAT FIRST POINT. THEN I WENT BACK AND CHKED THE DEP IN THE FMC WHICH THE FO HAD BUILT. THE RTE WAS BUILT TO FLY DIRECT CRI THEN OUTBOUND TO A POINT 50 NM ON THE OTHER SIDE OF CRI. I VISUALLY LOOKED AT THE RTE DRAWN AND IT LOOKED CORRECT AS DISPLAYED ON THE HSI. WITH THE KENNEDY VOR DIALED IN AND THE 223 DEG RADIAL SET ON MY VOR TUNER I WAS FINISHED. THE FO STATED HE ALSO HAD THE DEP VOR AND RADIAL SET IN HIS VOR TUNER. I DID NOT NOTICE THAT HE HAD ENTERED 176 DEGS IN THE VOR INSTEAD OF 223 DEGS AND HAD DONE THE SAME IN THE FMC. THE 176 DEG IS USED FOR THE 'CRI.' CLB OUT. ON CLB OUT; FLYING THE BUILT LNAV DEP; WE WERE FLYING OUT THE WRONG RADIAL. ATC ADVISED US TO INCREASE OUR RATE OF CLB AND SHORTLY AFTERWARDS WE HAD AN RA ON THE TCASII. WE HAD VISUAL CONTACT ON THAT ACFT AND PASSED ABOVE THE ACFT. ATC THEN VECTORED US TO THE R TO INTERCEPT THE 223 DEG RADIAL. THE CTLR TOLD US TO CALL NY DEP UPON LNDG. I CALLED THEM AND THEY SAID A PLTDEV WOULD BE FILED. SUPPLEMENTAL INFO FROM ACN 367310: WHEN I WAS PROGRAMMING THE CDU FOR THE BREEZY POINT CLB; I INCORRECTLY PUT THE 176 DEG RADIAL OUT OF CRI INSTEAD OF THE 223 DEG RADIAL. I BACKED UP THE 176 DEG RADIAL ON THE VOR SO THAT THE 2 RADIALS AGREED. WHEN I WAS READING THE BREEZY POINT CLB PROCS I LOOKED DOWN AT THE CANARSIE CLB BY MISTAKE AND USED THE WRONG RADIAL. MY BACKUP WAS THE WRONG RADIAL SO THE WRONG RADIALS MATCHED.