NMAC BTWN 2 DEP ACFT SAME DIRECTION TFC.

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LEVEL 12000; NY APCH CALLED ACR MLG TFC; 8 O'CLOCK (OR SO) 10 MI; CLBING TO 11000. WE RESPONDED 'NO JOY.' THEN ATC CALLED MLG 6 MI; 11000. WE AGAIN RESPONDED 'NO JOY.' THEN ATC TOLD US 'THE MLG HAD US IN SIGHT; WAS MAINTAINING VISUAL; CLBING TO 17000. NO PROBLEM.' SHORTLY THEREAFTER; I LOOK OUT THE L WINDOW; AND THERE GOES THE MLG ALMOST DIRECTLY OVERHEAD; AND QUITE CLOSE. I ASKED ATC WHAT THE VERT SEPARATION WAS; AND THE ANSWER WAS '400 FT.' (MLG CAME FROM BEHIND -- OVERTOOK US). I TURNED R ABOUT 45 DEGS TO AVOID ANY WAKE TURB; AND TOLD ATC AS MUCH. THEY SAID 'APPROVED.' FILED NEAR MISS THE NEXT DAY. I SINCERELY HOPE THAT THE OTHER ACFT WASN'T TRYING TO GIVE US A SCARE; BECAUSE THAT COULD COST THEM THEIR JOBS. SUPPLEMENTAL INFO FROM ACN 200299. AFTER HAVING DEPARTED RWY 4 AT LGA; WE WERE GIVEN A HDG OF 270 DEGS AND CLB TO 10000. AT 10000 FT AND 250 KTS WE WERE CLRED TO 11000 FT. TFC WAS CALLED AT OUR 1 O'CLOCK AT 12000 -- AN ENGINED TURBO PROP MDT. THE CAPT SAW THE TFC AND ACKNOWLEDGED IT 5-6 MI AWAY. I WAS HAND FLYING AND CONCENTRATING ON THE LEVEL OFF AT 11000 FT. I DECIDED TO ENGAGE THE AUTOPLT SO I COULD SCAN OUTSIDE EASIER. I THEN VISUALLY ACQUIRED THE TFC. DEP THEN TOLD US TO MAINTAIN VISUAL SEPARATION WITH THE MDT AND CLB TO 15000. I SIMULTANEOUSLY BEGAN AN ACCELERATION TO 300 KTS AND USING THE VS MODE COMMANDED A CLB TO 15000 AT 2000 FPM. THE CLB AND ACCELERATION WERE SMOOTH; BUT SLOW. THE MDT INITIALLY APPEARED TO BE MOVING R TO L AND I JUDGED WE WOULD PASS BEHIND THE ACFT DURING THE CLB. THE MDT WAS NOT ON OUR FREQ. THEN; EITHER THE MDT TURNED FROM SW TO W OR MY INITIAL PERCEPTION OF THE DIRECTION WAS INCORRECT. I MADE THE COMMENT TO THE CAPT 'THIS LOOKS TOO CLOSE' SO I MADE A 10 DEG L CORRECTION; CONFUSED AS TO WHAT 'MAINTAIN VISUAL SEPARATION' REALLY ALLOWED ME TO DO. COULD I TURN LEGALLY; WAS THERE ANOTHER ACFT TO MY L; HOW DID WE OVERTAKE THIS ACFT SO RAPIDLY? I IMMEDIATELY COMMANDED 2500 FT UP ON VS. WE PASSED L OF THE MDT LESS THAN 1000 FT AND ABOUT 6-800 FT ABOVE. WE WERE ON PARALLEL HDGS.