ACFT WITH SIMILAR CALL SIGNS (TWA FLTS 843H & 834; BOEING 747 & BOEING 727) APCHD JFK INTL ARPT FOR LDG. FLT 843H WAS 22MIN LATE & FLT 834 WAS 12 MIN EARLY, CAUSING SAME ARR TIME. NY TRACON VCTRD FLT 834 BHND FLT 843H. WHILE CTLG THE ACFT, JFK FINAL CTLR, INADVERTENTLY TRANSPOSED FLT NUMBERS & GAVE A VCTR FOR FLT '834H' (INTENDED FOR FLT 843H) TO TURN FINAL FOR A VOR RWY 13L APCH. FLT 843H DIDN'T READILY DETECT THE ERROR & CONTD ITS LAST ASSIGNED HDG OF 300 DEG. ABT 72 SEC LTR, FLT 843H QUERIED THE CTLR, BUT BY THAT TIME, THE CTLR NOTED HIS ERROR & GAVE FLT 843H A TURN TO REVERSE DRCTN. HOW-EVER, FLT 843H HAD ENTERED LAGUARDIA AIRSPACE & CONVERGED ON ANOTHER ACFT (PAN AM FLT 537), BOTH AT 3000'. FLT 537 WAS BEING VECTORED FOR AN ILS RWY 4 APCH TO LAGUARDIA ON ANOTHER FREQ. NOTING THE ACFT WERE CONVERGING, THE CTLRS TOLD EACH RESPECTIVE FLT TO CLIMB (WHILE THEY ATMTD TO RESOLVE THE PRBLM). FLT 537 CAME OUT OF CLDS; THE CREW SAW THE OTR ACFT, TOOK EVASIVE ACTN & PASSED WITH ABT 600' SEPN. THE JFK CTLR HAD 20/80 VISION, BUT WASN'T WEARING GLASSES AS REQUIRED.