RAMP OP PUSHBACK INSTRUCTIONS FROM RAMP TWR OPERATOR NOT FOLLOWED.

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ACFT FULL; LOTS OF EXCESS BAGS; A FEW MINS LATE TO DEPART GATE. AFTER REQUESTING PUSHBACK; RAMP TWR TOLD US TO WAIT FOR DC-9 IN ALLEY AND THEN CLRED TO PUSHBACK. I RELAYED THIS INFO TO TUG DRIVER VIA INTERCOM. WHILE WAITING; I TOOK THE OPPORTUNITY TO EXPLAIN THE DELAY AND WELCOME EVERYONE ONBOARD VIA THE PA. WHILE I WAS MAKING THE PA; THE RAMP TWR ISSUED AN ADDITIONAL RESTR TO WAIT FOR A FLT INBOUND AND THEN CLR TO PUSH. THE FO RELAYED THESE INSTRUCTIONS TO THE TUG DRIVER AND ASSUMED HE COPIED. A FEW MOMENTS LATER THE TUG DRIVER BEGAN THE PUSH. I ENDED THE PA WHEN THE ACFT BEGAN TO MOVE AND AGAIN MONITORED THE INTERCOM. AS WE WERE COMPLETING THE PUSHBACK; THE RAMP TWR BECAME VERY UPSET BECAUSE WE PUSHED PRIOR TO THE INBOUND ACR FLT AND GND CTL HAD TO REPOS THE FLT. WE VOLUNTEERED TO PULL BACK INTO THE GATE BUT THE RAMP TWR SAID IT WAS TOO LATE. THE TUG DRIVER SAID HE HAD NOT HEARD THE ADDITIONAL RESTR. SAFETY WAS NOT AN ISSUE; BUT OPTIMUM TFC FLOW WAS IMPACTED. MORAL: NEVER ASSUME ANYTHING. ALSO; I SHOULD HAVE CONTINUED TO MONITOR THE FREQS AS A BACKUP WHILE I GAVE THE PA.