transmission and gearbox, drive-shaft and prop-shaft

what are the difference with transmission and gearbox, drive-shaft and prop-shaft.

The transmission is housed within the gear box. When speaking of the shaft from the gear box to the rear differential, ‘drive shaft’ and ‘prop shaft’ are synonymous.

Thanks a lot for your help.

As you said, Drive shaft and Prop shaft are synonymous, why people created two different names for one product?

Is the difference just looks like american enlgish and britain english?

Possibly, but multiple names for things are certainly not limited to automotives or even to English. It happens in all fields in all languages.

Thank you for your kindly explanation, Mr Micawber