AN A320 CAPT CONTINUED AN IMC ILS APCH WITH A 20 KT AIRSPEED DIFFERENCE BETWEEN THE CAPT'S; FO'S AND STBY AIRSPEED INDICATORS.

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ON FINAL APCH WITH FLAPS 2 DEGS; WE RECEIVED AN OVERSPD WARNING. LOOKING AT MY AIRSPD INDICATOR; I SAW THAT WE WERE ON SPD AND AT THE CORRECT AIRSPD. THE FO'S AIRSPD INDICATOR WAS SHOWING 20 KTS FASTER AND THUS THE OVERSPD WARNING. XCHKED THE STANDBY AIRSPD INDICATOR AND IT WAS CLOSER TO THE FO'S AIRSPD. I CLICKED OFF THE AUTOPLT AND AUTOTHROTTLES TO MAINTAIN AIRSPD BTWN MINIMUM AND MAX; WHICH WASN'T THAT LARGE A MARGIN SINCE WE WERE ALREADY CONFIGURING. WE CONTINUED WITH THE APCH; BECAUSE ALTHOUGH IN RAIN AND IMC TO ABOUT 600 FT; I FELT THAT WAS A SAFER COURSE OF ACTION THAN TRYING TO GO MISSED APCH IN HVY RAIN IMC WHEN WE WERE MORE ON A STABILIZED APCH AT THAT POINT. THE REST OF THE APCH WAS UNEVENTFUL WITH NO ADDITIONAL OVERSPD/UNDERSPD WARNINGS AND LANDED APPROX 2000 FT DOWN RWY XXR IN ZZZ. DEBRIEFED MAINT ONCE WE ARRIVED AT THE GATE OF THE PROB. SUPPLEMENTAL INFO FROM ACN 795835: I BELIEVE THERE WAS A HUMAN FACTORS ISSUE OF PRESSING ON. IN OTHER WORDS OF TRYING TO COMPLETE THE MISSION THAT WE WERE ON INSTEAD OF BREAKING OFF THE APCH WHEN THE DISCREPANCY WAS FIRST NOTICED. I THINK THE APCH SHOULD HAVE BEEN DISCONTINUED AND SOME TIME TAKEN IN A LESS TASK SATURATED PHASE OF FLT TO PROPERLY ASCERTAIN THE AIRSPD DISCREPANCY AND THEN FLYING THE APCH WITH THE FO AS THE PF INSTEAD OF THE CAPT TRYING TO XCHK THE AIRSPD CONSTANTLY. ALSO; I BELIEVE THE APCH SPD WAS FLOWN 5-9 KTS FAST AS HE WAS FIGHTING THE TENDENCY NOT TO GET TOO SLOW EVEN THOUGH HIS AIRPSD INDICATOR READ ERRONEOUSLY SLOW.