ASTRA JET FLC FLEW TOWARD THE WRONG FIX DUE TO 2 FIXES WITH THE SAME NAME AND ATC ONLY PROVIDED THE NAME AND NOT THE IDENTIFIER.

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AFTER DEPARTING DEER VALLEY; AZ; AND ESTABLISHED IN THE CRUISE PORTION OF OUR FLT; WE WERE GIVEN A CLRNC TO FLY DIRECT WINSLOW; THEN ZUNI; THEN DIRECT DEST (COLUMBUS; OH). BEFORE REACHING WINSLOW OUR CLRNC WAS CHANGED TO FLY TO WINSLOW; THEN FARMINGTON; THEN DEST. FARMINGTON WAS NOT PART OF OUR FILED FLT PLAN; OR OUR INITIAL CLRNC; AND THE CTLR DID NOT FOLLOW FARMINGTON WITH ITS IDENTIFIER. AFTER A BRIEF DISCUSSION IN THE COCKPIT OF THE IDENTIFIER FOR FARMINGTON; 'FGT' WAS PUT IN THE FMC. A NORTHEASTERLY DIRECTION; CORRECT FOR OUR FLT; WAS DISPLAYED AND THE CAPT PROCEEDED IN LONG RANGE NAV. AFTER WINSLOW A TURN WAS MADE TO FARMINGTON 'FGT.' APPROX 15 MINS LATER ATC CALLED US WITH A 30 DEG TURN TO THE L AND CLB TO FL370. THE CAPT COMPLIED WITH THE CTLR'S INSTRUCTIONS AND WE IMMEDIATELY MADE VISUAL CONTACT WITH THE OTHER ACFT. AT THE SAME TIME THE CTLR GAVE THE OTHER ACFT SIMILAR AVOIDANCE INSTRUCTIONS. THE CAPT MAINTAINED VISUAL SEPARATION FROM THE OTHER ACFT AT ALL TIMES. WE WERE THEN TOLD WE WERE SUPPOSED TO BE NAVING TO FARMINGTON 'FMN.' WE WERE THEN GIVEN CLRNC TO DEST AND ASKED TO CALL ZAB. WITH 2 VORS OF THE SAME NAME; HAD WE QUESTIONED THE CTLR FOR A MORE DESCRIPTIVE CLRNC; OR HAD THEY INCLUDED THE IDENTIFIER WITH THE CLRNC; I BELIEVE THIS SIT COULD HAVE BEEN AVOIDED.