The difference in current between the two is going to be completely dwarfed by the power actually being supplied, since it is a power supply. Both ISL21070 and LM4040 are specified at tens of uA.
I was wondering about the difference in general, not just in a power supply. At first I had the same thought, that a few uA doesn't matter even in a low-power device, but then I thought about it some more. Let me know if my thoughts are wrong.
1. It could be hard to get a shunt voltage reference like the LM4040 down to its minimum current draw because you have to make a close estimate of the load and choose the bias resistor precisely.
2. If the load on Vref is shut off (e.g. a microcontroller goes into sleep mode and shuts off its ADC), the shunt reference will continue to draw the same current, except now it's all going through the shunt. On the other hand, the ISL21070 will only draw its quiescent microamps.
So maybe the difference in power consumption would matter in a small battery-powered device.