Difference between 7805 and ref02 module
I was wondering why a LM7805 (maybe $.50) wouldn't function in the
  same way?


Functionality is the same (apart from the trim feature on the REF02) but not performance....

The REF02 has an output voltage specified at 5V ± 0.2%

The 7805 is specified by Fairchild as 5V ± 4%

How much accuracy do you need?


The output voltage temperature stability of the REF02 is 10ppm / degC. This is equivalent to 50 uV / degC.

The 7805 equivalent spec is 5 mV / degC

How much stability do you want?


The long-term stability of the REF02 is ± 100 ppm (0.5 mV) over the first 1000 hours and gets better for the 2nd 1000 hours. I've not seen this specified for a 78XX regulator so it's not really guaranteed.

Horses for courses.