NMAC BTWN A C560 AND AN UNKNOWN ACFT ON SHORT FINAL AT SFB; FL.

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FLYING FROM ORLANDO (MCO) TO SANFORD (SFB) IN THE LATE AFTERNOON THERE WERE SEVERAL CELLS ALONG THE RTE. WITH SLIGHT DEVS AND AN EARLY DSCNT FROM 6000 FT MOST WX WAS AVOIDED. PRIOR TO BEING HANDED OFF FROM ORLANDO APCH TO SANFORD TWR WE RPTED THE FIELD (SFB) IN SIGHT AND WERE SUBSEQUENTLY CLRED THE VISUAL TO FOLLOW ANOTHER CITATION JET ABOUT 2 MI AHEAD. WE RPTED THE ACFT IN SIGHT AND WERE TOLD TO FOLLOW WITH S-TURNS APPROVED TO THE N IF NECESSARY; CONTACT TWR 120.30. WE IMMEDIATELY BEGAN TO S-TURN N AND SLOWED TO OUR FINAL APCH SPD (ORLANDO APCH ALSO REQUESTED THIS) TO GET ADEQUATE SPACING AS WE MONITORED TWR. THE FREQ WAS HEAVILY CONGESTED SO WE CONTINUED OUR S-TURNS AND THE VISUAL WE WERE CLRED. PROBABLY ABOUT 1 MI FROM THE APCH END OF RWY 9L WE RECEIVED AN RA TO DSND; DSND. WE ACQUIRED TFC VISUALLY AND CONTINUED OUR VISUAL. IF WE WOULD HAVE CONTINUED ON OUR PREVIOUS COURSE AND NOT DSNDED WE WOULD HAVE PASSED WITHIN AN ESTIMATED 100-200 FT. AT THE TIME OF OUR RA TWR TOLD THE OTHER ACFT TO START AN IMMEDIATE CLBING R TURN AND THAT SHE HAD NO IDEA WHAT WE WERE DOING THERE BECAUSE NOBODY WAS TALKING TO US. THE OTHER ACFT RPTED US IN SIGHT. I IMMEDIATELY KEYED UP AND SAID WE WERE 'CALL SIGN; CLRED FOR A VISUAL BY APCH; S-TURN TO THE N IF NECESSARY.' SHE CLRED US TO LAND. ONCE ON THE GND TWR CONTACTED US BY PHONE TO HEAR OUR RPT. WE EXPLAINED TO THEM THAT APCH CLRED US FOR THE VISUAL BEHIND THE CITATION S-TURN TO THE N IF NECESSARY. IT KIND OF SOUNDED LIKE THEY NEVER KNEW WE WERE THERE. APCH FORGOT TO LET THEM KNOW WE WERE THERE. ALSO THE HVY TSTM APCHING THE AREA ALONG WITH A BUSY FREQ/PATTERN AT SANFORD LED TO A NEAR MISS.