Oils and lubricants are a vital part of any mechanical assembly on your car, at least as much so as any connecting rod, bearing or whatever else you care to name. The liquid components of your transmission and differential are as carefully tailored to their tasks as any other part; while they sometimes can interchange, there's simply no substitute for using the fluid for which your transmission or gearbox was designed.
Manual gearboxes and automatic transmissions operate on different but related principles. The manual gearbox is just that: a box containing two shafts with inter-meshing gears mounted on them. An automatic transmission also contains a gearset, but it uses a "planetary" gearset mounted at the rear. A set of clutches control planetary gear engagement, either spinning them by locking them to the transmission's input shaft or holding them still by locking them to the case. The clutch packs get their locking power from high-pressure fluid supplied by the transmission's oil pump.
Basic gearbox oils are almost identical to motor oil -- they even look like it. The primary difference between the two is that gearbox oils are very thick; the average modern gearbox uses a 75W-90 weight oil, compared to the 5W-30 to 10W-40 used for motor oil. Contrary to popular belief, gearbox oils aren't typically formulated to be as slippery as possible, the way that differential oils are; that would interfere with the transmission's synchronizers, which need a bit of friction to work. Rather, the gearbox oil's main priority is to provide a soft cushion that prevents shock damage to the gears and resist breaking down under constant shear forces.
Transmission fluid is a very thin sort of hydraulic oil -- typically around 0W-5 to 5W-10 weight. Transmission fluid needs to be very thin so it can quickly and efficiently flow through the hundreds of tiny fluid channels in the average automatic transmission. Planetary gearsets rely on the fluid for lubrication just like the standard gearbox does, but they're designed with tolerances and materials better suited for the thinner oil. Automatic transmission fluid also contains a number of friction modifiers designed to keep clutch engagement consistent through the life of the oil, as well as surfactants -- detergents -- to keep the channels free of blockage, and anti-foaming agents to keep air out of the hydraulic system.
While ATF and gearbox oil are very specialized, there actually is a bit of room for interchange between the two. Several kinds of manual gearbox use automatic transmission fluid, notably the Borg-Warner T-5 used in everything from Ford Mustangs to import minivans. The T-5 and gearboxes like it can use standard ATF because -- like the automatic -- they were designed specifically for it. The clearances between the gears, the materials used and the design of the synchronizers were set from day one to use ATF -- so you can use it in certain transmissions. Do not, for any reason, use gearbox oil in an automatic -- at least, not if you intend to drive the car more than three blocks. Ever.
