overdrive button on automatic gearbox was introduced by not sure european or japanese car maker last time .
the reaosn they put overdrive button on gearbox is to turn the 4th or5th gear , the top gear on or off .
why ? becasue during city driving , automatic gearbox is very sensitive , they judge the gear selection by the speed and throttle , 20-30 , it will run at 2nd , 40-50rd , 60 upwards can use 4th gear already . during city driving , there are many stop and go , the gearbox dont know what we are goign to stop or go , and very often , reach 60km/h time , u suddenly need to brake slow down and then press again , and the gear will shift between 3rd and 4th gear , which causes gearbox jerk . that is why OD button was introduced in automatic car which is actually to control the top gear to eliminate shift changes .

if you are driving a manual gear car , u can see the surrounding , sometimes at 60 , if you know u want to slow down or what u wont change to a higher gear because u know soon or later u need to downshift again, if u do so . but an automatic gearbox isn't that smart . they will just keep on changing as long as the speed is increasing .

yeap . that was the idea of the OD button . it is for city driving , reducing the gear change jerk .
as overtaking , it doesn't mean that way , becasue overtaking u can just floor ur pedal .

as i said the OD is like the top gear . it will go into the top gear if it passes the power range .
when u floor it , the ECU or sensor will detect it as a sudden power , hence it will downshift to give more torque .

the problem is in city driving , u wont be pressing the pedal hard or what . suddenly u are doing 60 , and it will shift into 4th . just after u hit 60, u suddenly see the car in front slow down , u will slow down also . and the gear will not downshift so fast , hence when u want to accelerate , it will dragg 50km/h at 4th gear(top gear) , ur car will feel sluggish . then only u floor it to make it downshift to 3rd gear to accelerate back to 60 .
but in manual , u can control better , that is why OD is enable in auto boxes .

well , i am not quite sure about manual type ..
i used to drive a mitsubishi tredia last time , drove once ...
it has 4 forward gear , and there is another gear stalk , which engage OD .

OD = last gear .

enough said.