NMAC. CONFLICTING TFC RPTED BY ATC; BUT FLC IDENTED IT AS ANOTHER ACFT. CLRED TO CLB; BUT MAINTAIN VISUAL SEPARATION; FLC REALIZED THE PREVIOUSLY IDENTED TFC WAS NOT THE CONFLICTING TFC; RECEIVED TCASII AND TOOK EVASIVE ACTION.

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AFTER DEPARTING BUR RWY 15 WE WERE ASSIGNED ABOUT A 340 DEG HDG AND 6000 FT. (MIGHT HAVE BEEN 5000 FT). DURING THE R TURN A SKYLANE (CESSNA) WAS GIVEN AS TFC. THE FO SAID WE HAD THE TFC IN SIGHT AT 2 O'CLOCK AND 4 OR 5 MI. I COULD NOT SEE THE SKYLANE DUE TO THE R BANK TURN. THE SKYLANE WAS VFR 500 FT ABOVE OUR ALT. AFTER I RPTED THE TFC IN SIGHT; (I RPTED TFC IN SIGHT BECAUSE FO SAID HE SAW THE TFC); WE WERE GIVEN PLT'S DISCRETION TO CLB TO 13000 FT WHILE MAINTAINING VISUAL SEPARATION FROM THE SKYLANE. JUST AFTER THE WINGS WERE LEVELED I SAW THE SKYLANE. THEN THE FO STARTED TO CLB USING THE VERT SPD. THIS LOOKED TO ME LIKE WE MIGHT HAVE A TFC CONFLICT BY CLBING AND STATED SO TO THE FO. AS THE SIT LOOKED WORSE IN A SHORT TIME I STARTED PUSHING THE NOSE BACK DOWN TO GO BENEATH THE SKYLANE. AT THAT SAME TIME TCASII SOUNDED A 'DSND' RA. I THINK IT WAS AT THAT TIME WE BOTH REALIZED THAT THE FO HAD BEEN LOOKING AT A JET FURTHER AWAY; THINKING HE WAS LOOKING AT THE SKYLANE TFC. A MISIDENT OF THE TFC ISSUED TO US. IT DID NOT APPEAR THAT THE SKYLANE TOOK ANY EVASIVE ACTION. WHILE IT IS NOT ALWAYS PRACTICAL; BOTH PLTS SHOULD BE IN AGREEMENT THAT THE CORRECT TFC IS IN SIGHT.