Zener Diode vs Precision Voltage Shunt
If you look at an average zener diode like the MM3Z3V6T1G, and a voltage regulator like the LP2980,  you can see the difference pretty quickly.

First off, the zener voltage is listed as 3.4V-3.8V. That's already +-5%.

Next, you have the IV curve of the zener.



Notice how the Zener breakdown curve on the left is not a straight vertical line, but is instead somewhat diagonal. The gradient of this line is the zener resistance. In the attached datasheet, it lists the resistance as 90 Ohms (at 5mA). This means that the voltage will vary depending on the current through it, by almost 100mV per mA. Say you want to use it at 0mA and 3 mA, thats a variance of another 8%.

In contrast, the regulator is guaranteed to have an absolute max variance of +-2.5%, and keep that constant between 0mA and 50mA. These are maximums, and the actual value will likely be a fair amount lower.

Basically, with Zeners you're almost guaranteed to get variations of +-10%, whereas with actual regulators its more like &lt;1%.

Just to add, Andy is correct that this is a voltage regulator, but the same logic applies to a shunt like the LT1389, except that it's even more accurate at a guaranteed +-0.5%.