CL60 CREW HAD A TRACK DEV IN ZAU CLASS A.

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PLT FAILURE TO FOLLOW ATC INSTRUCTION. FLYING AT FL350 FOLLOWING SEVERAL ATC VECTORS FOR CLB FROM MDW; WE LEVELED AT FL350. A LINE OF TSTMS WERE ALONG AND N OF OUR RTE AND OUR VISION WAS OBSTRUCTED BY CIRRUS AND SOME 'BLOW-OFF' FROM TSTMS S OF OUR RTE. WE ASKED FOR CLB TO FL390. CTR GAVE US A 'TURN R 10 DEGS CLB TO FL390; CONTACT CHICAGO ON 1XX.XX.' COPLT READ BACK 'R TURN 10 DEGS;' AND THE BAL; I SET IN A 10 DEG L TURN (S OF TRACK). WHEN COPLT CONTACTED NEXT CTLR HE SAID; 'ACFT X LEAVING FL350 FOR FL390 HEADING 275 DEGS.' NEW CTLR SAID 'WHAT? WHAT IS YOUR HEADING?' COPLT REPLIED WE SET IN WRONG HEADING. I BEGAN TURNING 20 DEGS R (N OF TRACK). CTLR TURNED AN ACR Y FLT 10 DEGS R AND CANCELED HIS DSCNT CLRNC; THEN CONFIRMED OUR NEW HEADING AND ADDED ADDITIONAL 10 DEGS R; THEN CHANGED OUR CLB LIMIT TO FL370. AFTER ACR Y PASSED 8-10 MI S OF US WE WERE CLRED TO FL390. THIS WAS MY FIRST TRIP WITH ONE OF OUR NEWLY HIRED PLTS AND WAS THE THIRD LEG OF A 2 DAY TRIP. HE THOUGHT I HAD HEARD SOMETHING OTHER THAN HE HAD; BUT HAD READ BACK THE R TURN; AND I HAD HEARD 'R TURN 10 DEGS' AND HAD WANTED TO BE S OF COURSE; BECAUSE OF WX; SO 10 DEGS R BECAME 10 DEGS S AWAY FROM WX. HE WAS ALSO JUST ABOUT TO QUESTION ME WHEN THE CTLR ASKED 'WHAT...' THE CL604 OPERATES ON 1/2 BANK FUNCTION ABOVE FL315/FL320 BY DEFAULT; THUS HEADING CHANGES ARE SLOWER THAN EXPECTED BY THE CTLR (SUPPOSITION ON MY PART) AND RESULT IN A DELAY IN COURSE/TRACK CHANGE. I BELIEVE MY ACTIONS WOULD HAVE BEEN CHALLENGED EARLIER WITH AN ESTABLISHED CREW MEMBER; HOWEVER MY NEWER COPLT WAS FORMULATING A TACTFUL QUESTIONING OF MY ERROR.