AT APRX 2211 CDT, DELTA FLT 314 (BOEING 727, N482DA) WAS CLRD TO TAXI TO RWY 31 WITH INFO INDIA. INFO INDIA INDCD RWY 31WAS ACTIVE. SOON THEREAFTER, SOUTHWEST FLT 715 (BOEING 737, N50SW) REQD & RECD TAXI CLNC TO RWY 13 FOR TKOF. TO ACCOMMO-DATE INCOMING TRAFFIC THAT HAD REQD RWY 13 FOR LNDG, FLT 715 (AT THE APCH END OF RWY 13) WAS CLRD FOR IMMEDIATE TKOF. INITIAL ACKNOWLEDGEMENT (OF TKOF CLNC, RECD BY TWR) WAS GARBLED FLWD BY AN AUDIBLE '. . . 715.' THE TWR CTLR DID NOT REALIZE BOTH FLTS HAD ACKNOWLEDGED THE TKOF CLNC, SINCE FLT 314 WAS STILL TAXIING TO RWY 31. BOTH FLTS STARTED DEPG FM OPPOSITE ENDS OF THE RWY. AT 1ST, NEITHER FLT CREW COULD SEE THE OTR DUE TO A CROWN IN THE RWY. WHEN THEY SAW EACH OTR, FLT 715 WAS APCHG V1 SPEED & ELECTED TO CONT THE TKOF. FLT 314 HAD JUST BGN ITS TKOF ROLL & ABORTED BY TURNING OFF ON A TWY. AN INV REVEALED THAT BFR DEPG, WHEN FLT 314 WAS APCHG RWY 31 FOR TKOF, THE 1ST OFFICER (F/O) THOUGHT HE HEARD CLNC FOR TKOF. THE CAPT OF FLT 314 ASKED THE F/O IF HE WAS SURE & THE F/O REPLIED 'YES' & SAID HE HAD ACKNOWLEDGED THE CLNC.