LGT MAKES MISSED APCH AND HAS TCASII RA.

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THIS EVENT OCCURRED APPROX 20 MI NW OF RWY 12 AT MIA. WE WERE AT 3000 FT MSL AND APPROX 200 KIAS. WE WERE GIVEN A DOGLEG VECTOR TO FINAL FOR RWY 12. DUE TO A STRONG TAILWIND AND SEVERE WX ON FINAL APCH; CAPT ELECTED TO BREAK OFF THE APCH. WE WERE GIVEN A R TURN TO A NORTHERLY HDG AND CLRED TO CLB TO 6000 FT. AFTER COMPLETING THE TURN AND WHILE CLBING AT APPROX 1000 FPM; WE NOTICED TFC AT 11 O'CLOCK HIGH. WE IMMEDIATELY CUT THE CLB RATE AND SIMULTANEOUSLY GOT A 'DSND' RA ON TCASII. THIS OCCURRED AT APPROX 5600 FT. WE FOLLOWED THE DSND RA AND WHILE IN THE DSCNT; AT APPROX 5300 FT; ATC ADVISED US TO 'DSND AND MAINTAIN 5000.' WE LEVELED OFF AT 5000 FT AND THE TFC PASSED OVERHEAD WELL CLR. THERE WERE NUMEROUS STRAY TSTMS IN THE AREA AT THE TIME AND A HVY LOAD OF JET AND TURBOPROP TFC. THE CTLR WAS VERY BUSY. A CONSTANT PROB WITH SEVERE WX IN TERMINAL AREAS IS THAT THE CTLRS DO NOT SEE THE SAME LEVEL OF TSTM ACTIVITY THE PLTS DO. A DANGEROUS RESULT IF THE SIT IS THAT PLTS DO NOT AND CANNOT ALWAYS FOLLOW CTLRS' INSTRUCTIONS OR THE APCH CLRNCS.