What is the different between an aircraft engine gearbox and an aircraft engine transmission?
In aircraft, transmission usually refers to the whole system, that transmits power between two points, say between the MGB and the tail rotor. Gearbox is usually a component of the transmission system (along with shafts, for example) which is used for changing rotational speed.

In a sense, a full transmission system is found only in helicopters, and not in fixed wing aircraft. The closest thing a fixed wing aircraft has to a transmission is the variable pitch propeller, which is analagous to the car transmission.



By nydano - Cut away view of Hamilton Standard Propeller, CC BY-SA 2.0, $3

On the other hand, a number of aircraft have gearboxes, especially the turboprops (and some piston engines), for reducing the engine output speed before turning the propeller. The following image shows the gearbox in a GE turboshaft engine.



Image from mrlarry.org

Turbofan engines do have a gearbox, which is used to run the accessories. The compressor is usually run through a shaft from the turbine, with various turbine stages running various compressor stages, while some turbofans do use gears between turbine and compressor, called geared turbofans.

Aircraft transmissions (whether helicopter or fixed wing) is usually one speed from one point to another, with speed reduction (or direction change) offered by the gearboxes.