Half wave, as its name implies, only rectifies one half of the supply sine wave, whereas full wave results in both halves of one cycle being utilized. The full wave results in a higher energy content.
Max.
half wave rectification clips the negative cycle completely, causing the output to be at 0 volts for half of each cycle. Full wave rectification flips the negative side of the sine wave to the positive side. Full wave rectification is less wasteful.
Here you can see what we are referring to:
The capacitors that are often added on after a rectifier are to smooth the signal to make it look more like a constant DC voltage.
Another consideration is that, if rectifying a power transformer output, a half-wave rectifier has a DC current component which can saturate the transformer core and cause high transformer primary currents and transformer overheating, so the transformer must be significantly derated for use with a half-wave rectifier.