FLC DEPARTS FROM ASSIGNED ALT.

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CE550. DEP 2 CREW; 2 PAX FROM HPN. ASSIGNED HDG OF 220 DEGS AND LEVEL AT 12000 FT. NEW YORK DEP THEN ASSIGNED HDG OF 240 DEGS. THE PIC; PF AT TIME; TURNED TO HDG OF 240 DEGS. THE FO READ BACK 'UP TO 240' AND SET THE ALT ALERTER. 1 PAX WAS ALSO FORWARD ASKING FOR ADDITIONAL HEAT IN PAX COMPARTMENT. MOMENTS LATER THE FO POINTED TO THE ALT ALERTER SUGGESTING I START MY CLB TO FL240. AS THE CLB WAS STARTED I REALIZED THAT HE (FO) HEARD AN ALT CLB TO FL240; AND I HAD ONLY HEARD A TURN TO A HDG OF 240 DEGS. ATC WAS IMMEDIATELY QUERIED FOR A CONFIRMATION. OUR CLB WAS STOPPED AT 12800 FT. NEW YORK DEP VERIFIED A TURN ONLY TO HDG 240 DEGS AND TO STOP CLB AND MAINTAIN 12800 FT. THERE ARE SEVERAL FACTORS CONTRIBUTING TO THIS EVENT. 1) PIC SHOULD ALWAYS VERIFY HDG AND ALTS WITH THOSE SET AND READ BACK BY FO. IN THIS EVENT I WOULD HAVE NOTICED HIS HDG BUG STILL SET AT PREVIOUSLY ASSIGNED HDG OF 210 DEGS AND ALT ALERTER SET AT FL240. A CONFLICT REQUIRING IMMEDIATE VERIFICATION. 2) LESS INTERFERENCE FROM PAX ESPECIALLY DURING PERIODS OF FLT OVER HIGH DENSITY AIRSPACE.