Why use a Zener in a regulator as opposed to a regular diode?
You could make a 5.6 volt diode with 8 forward connected diodes that each have a forward voltage of 0.7 volts at 1 mA. Ok that's one point on the graph and at room temperature with 5.6 volts applied you would get 1 mA.

If you lowered the voltage to 0.6 volts across each diode (4.8 volts in total), the current would reduce approximately ten times to about 0.1 mA.

And this is where the comparison breaks down. The same percent reduction in applied voltage across a zener would result in a significantly lower current being taken.

In other words, a zener has a much sharper knee characteristic. Same story if increasing the voltage; the diodes wouldn't conduct as much as the zener.

So, the zener has a much sharper knee characteristic compared to a bunch of series connected diodes.

Zener can also be made to have a fairly neutral temperature stability whereas diodes have a recognized voltage-temperature gradient that is almost etched in stone (all due respect to Moses).