I don't really understand how a ball that can only control three axis, is any better than a joystick that can only control three axis...?
Well, it isn't. But a ball can be more intuitively made to control the spacecraft on the necessary six
degrees of freedom, IMHO.
Imagine gripping that ball, or a rock, or a potato. It's a firm grip, and your fingers probably sink into the ball (and may reach "hands on stick" buttons for further functions), but that doesn't make it different from a joystick yet. What does is the next step, where you imagine the ball is in fact the shuttle. (Heck, Type 7 even looks quite a bit like that!)
Push the ball forward without twisting and the shuttle moves forward. Twist the ball forward and the shuttle's nose tilts down. Now, you could build a joystick that does these same things (that is, accepts "push" as a valid command rather than just "twist"), and you could cover the sideways moves the same way. But with a grip of a ball, you can also lift
the control so that the shuttle goes straight up, which is not all that comfortable with the vertical shaft of a joystick. Plus you get a bit of extra leverage for twisting around the vertical axis.
The big thing there is that the ball (if soft on the surface) fits all hands. In order to twist or lift a joystick, you have to customize it for your own hand size before you get comfort and leverage.
As for hanging on to the controls, well, that works for the steering wheels of cars or boats, as those only cover one degree of freedom and are immune to stresses in other directions. Hanging on to a joystick or joyball for dear life will always send unwanted control input to your vehicle...
However, I was wondering if anyone knew what actual spacecraft, like the Space Shuttle or Soyuz, use for their control systems? I've been able to find pictures of their controls, but nothing that explains how they're used yet.
Good question. The shuttle controls for seated pilot and commander are obviously classic and trivial - a joystick for wing trailing edge control surfaces (pitch and roll) and pedals for rudder (yaw), and of course there's no throttle because the shuttle is a glider. But supposedly reentry is also initiated when seated, so the joysticks must have a secondary mode for orienting the vehicle with RCS burns before the OMS burn.
Dockings and the like are probably done with the aft console controls. There's a classic joystick there for the manipulator arm (frequently updated), and two sticks that look like automobile shifting sticks, supposedly giving two degrees of freedom each (looking much like the original eighties versions even in latest pics). No doubt there are dozens of modes for using those sticks to control the RCS thrusters in sequence, depending on what is to be achieved - and dozens more from all sorts of software modernizations and bypasses that depend on plugging in a laptop that's a hundred times more powerful than the vehicle's own computers.
But I guess an astronaut has no real control over individual nozzles, save perhaps for fiddling with the fuses.