Can overdrive give the same results?
Basically yes.
An overdrive usually gives a 30% reduction in engine speed at a given road speed. Or if you have the horsepower to do it, a 30% increase in speed.
The exact numbers depend on the gears in the OD so I'm just using 30% to illustrate.
The advantage as I see it of using an overdrive over changing the rear gears is that you then have the best of both. You have the lower gears that allow hauling a heavier load and the ability to run at lower RPM at speed.
If you change the rear gears you loose the leverage those lower gears represent.
I don't know why Crenweldge prefers an overdrive to changing gears, but I'm sure he has a reason and will come back and share.
I prefer a transmission that has an overdriven top gear. A 3-speed with an overdrive means you are shifting the 3-speed and managing the overdrive for best performance.
I have a 4-speed with an overdrive and find it annoys me to manage that overdrive. If I ever replace that trans a 5-speed or maybe a 6-speed will go in.
This same logic is why given the choice I'd choose a 4L80e over adding a Gear Vendors overdrive to a TH400. With the 4L80e you are not managing that overdrive the transmission and its control box are doing it for me.