FLC OF A BRITISH HAWKER 800 FAILED TO FOLLOW RWY HDG AFTER TKOF UNTIL GIVEN VECTOR BY DEP CTL CAUSING LTSS WITH ANOTHER ACFT.

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ACFT WAS ON A TWR-ENRTE IFR FLT PLAN FROM LAX ARPT TO BUR ARPT. I HAD BEEN INSIDE FBO ATTENDING TO FUELING MATTERS AND THE SIC HAD RECEIVED OUR ATC CLRNC. UPON ENTERING THE ACFT; I WAS INFORMED BY THE SIC WHAT CLRNC WAS AS WELL AS LOOKING TO SEE IF HE HAD PROGRAMMED IT INTO THE FMS. WE DISCUSSED THE CLRNC AND WERE THEN INSTRUCTED TO TAXI TO N COMPLEX RWY 24. ABOUT 1/4 OF THE WAY TO N SIDE; ATC ADVISED US TO TAXI BACK TO S SIDE RWY 25R. THEY THEN CHANGED IT TO RWY 25R AND CHANGED OUR ALT FROM 2000 FT TO 3000 FT FOR DEP. READY FOR DEP ON RWY 25R; INTXN DEP; I THEN DISCUSSED DEP CLRNC WITH SIC. WE BOTH AGREED; ALTHOUGH DIFFERENT; TO FLY RWY HDG TO LAX VOR AND THEN PROCEED VIA THE LAX 316 DEG RADIAL NWBOUND TO SILEX INTXN. WE DEPARTED AND FLEW TO LAX VOR AND I BEGAN TO MAKE R TURN TO INTERCEPT RADIAL/COURSE. UPON ROLLING WINGS LEVEL ON COURSE; I SPOTTED ANOTHER ACFT DEPARTING TOWARDS US OFF N RWY COMPLEX. I ASKED/VERIFIED WITH SIC IF THIS 'LOOKED RIGHT.' THE SIC ANSWERED 'YES.' I RESPONDED WITH 'I DON'T THINK THIS IS RIGHT.' I THEN EXECUTED A CLBING L TURN BACK TOWARDS A SW HDG AWAY FROM TFC. WHEN ACFT WAS IN L TURN; ATC INSTRUCTED US TO TURN L IMMEDIATELY. I THEN KNEW FOR SURE SOMETHING WAS NOT RIGHT. WE WERE TOLD TO FLY A 230 DEG HDG AND MAINTAIN 2000 FT. AT THAT TIME; WE THOUGHT ATC HAD MADE A MISTAKE. WE WERE NEVER ISSUED A PHONE NUMBER TO CALL UNTIL WE ARRIVED AT BUR ARPT. THEY TOLD ME (TWR MGR) THAT WE WERE TO MAINTAIN RWY HDG AND EXPECT RADAR VECTORS TO LAX 316 DEG RADIAL. SINCE THIS WAS A TWR-ENRTE CLRNC; ALL WE HAD WAS A SCRIBBLED DOWN CLRNC ON A PIECE OF PAPER THAT THE SIC HAD WRITTEN. WE HAD NO NORMAL GRAPHICAL INST DEP PROC TO LOOK AT. LAX DEP ATIS SAYS DON'T EVEN READ BACK CLRNC EXCEPT FOR XPONDER CODE AND CALL SIGN. I BELIEVE FOR SOME REASON WE WERE THINKING THAT WE WERE DEPARTING FROM N RWY COMPLEX. MULTIPLE RWY CHANGES AND ALT CHANGES ADDED CONFUSION. WE WERE ALSO POS AND HOLD AND I ACTUALLY DISCUSSED N RWY COMPLEX DEPS. WE HAD POSSIBLY FIGURED WE WERE WAITING FOR N SIDE TFC TO CLR.