The shower valve is the inner device within the wall that regulates water temperature, flow, and output to the showerhead, tub spout, and other shower accessories. The cartridge in the valve connects to the handle and has holes that regulate water flow and temperature when you turn the handle.
A stem is used in two or three handle shower valves, where one handle is responsible for cold water, another is responsible for hot water and the third would be a diverter stem that transfers water from the tub spout to the shower head. A cartridge is used in faucets to control hot and cold water through just one handle.
The shower balancing spool regulates the hot-to-cold water ratio, and equalizes the pressures to avoid extreme hot or cold water temperature fluctuations.