AFTER LOSING SIGHT OF THE RWY ON VISUAL APCH INTO SWF; A CL65 CREW INITIATE A GAR AND CONFLICT WITH AN INBOUND F100 WHILE BEING VECTORED FOR A SECOND APCH.

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ON APCH INTO SWF; ATIS RPTED CEILING 3000 FT; VISIBILITY 7 MI; AND ILS APCHS TO RWY 9. NEW YORK APCH GAVE US VECTORS FOR A VISUAL. WE ACQUIRED THE RWY AND CALLED IT IN SIGHT AND THEY CLRED US FOR A MISSED APCH. WE THEN CONTACTED THE TWR AND COMPLETED OUR BEFORE LNDG CHKLIST. AFTER A SHORT TIME; I LOST SIGHT OF THE RWY (TWR LATER SAID A LOW CLOUD LAYER HAD MOVED THROUGH). I ASKED MY FO (PNF) TO HAVE THE TWR TURN UP THE RWY LIGHTS. WHEN THEY WERE TURNED UP; WE COULD SEE WE WERE NOT IN A POS TO LAND. I INITIATED GAR. THE TWR CTLR GAVE US A HDG AND ALT THEN BROUGHT US AROUND FOR A TIGHT R VISUAL. I FELT THIS WAS SAFE AND ACCEPTABLE. WE COULD NOW SEE THE RWY AND LAND NORMALLY. WHILE WE WERE TURNING IN; ACR Y WAS TOLD TO SQUARE THEIR TURN TO FINAL. WHEN I TALKED TO THE TWR CTLR LATER; HE SAID HE THOUGHT THAT WOULD HAVE GIVEN US ENOUGH ROOM TO LAND. HOWEVER; ACR Y DID NOT SQUARE THEIR TURN AND WE COULD NOT SEE THEM DUE TO THE LOW CLOUD LAYER THAT WAS MOVING THROUGH. CONSEQUENTLY; WE GOT AN RA AND TWR GAVE US A TURN AWAY FROM FAC. I EXECUTED THE RA AND I THINK ACR Y EXECUTED ONE IN THE OPPOSITE DIRECTION. THE TCASII PROBABLY SAVED OUR LIVES. MY GUESS IS WE PROBABLY HAD ABOUT 600 FT OF VERT SEPARATION WHEN WE CROSSED PATHS. WHEN THE TWR PUTS ILS APCHS ON THE ATIS; THAT IS WHAT WE SHOULD BE VECTORED FOR. APCH ASKED AN ARRIVING ACFT IN FRONT OF US NUMEROUS TIMES IF HE HAD THE FIELD RIGHT UP TILL HE WAS PRETTY CLOSE. ACR Y WAS ALSO BEING VECTORED FOR A VISUAL APCH. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED HAD IT NOT BEEN FOR THE TCASII RA THEY WOULD HAVE COLLIDED WITH THE F100 ALSO ON APCH FOR THE SAME RWY. LATER; IN A DISCUSSION WITH THE TWR; HE WAS TOLD THE CTLR WORKING AT THE TIME WAS VISIBLY SHAKEN BY THE INCIDENT. RPTR WAS TOLD THAT THE TWR HAD ADVISED APCH CTLR ABOUT THE WX AND WERE ADVERTISING ILS APCHS ON THE ATIS; HOWEVER; THE TRACON HAD CONTINUED WITH VISUAL APCHS.