Slow down a little here - I am not saying that one or the other is quieter or noisier.
What I am saying is that a high impedance anything generates more noise than a lower impedance anything. (If and only if every other parameter remains the same.)
Don't jump to a conclusion that I did not make. It is easy to show that a MOS device with a small signal impedance of 50 ohms, has less noise generation that a 1Meg resistor. The opposite case can also be made.
Also, an amplifier that uses a cascode current source as an active load (and we all do that at one time or another in an op-amp) has a high impedance node looking into the active load. That generates a good amount of inherent noise. In a lot of cases, with mixed signal applications its not an important thing, because the signals you are working with are a lot bigger than the noise.
But, the discussion at hand was for a "very accurate bandgap" and that, in my mind means accurate nominal (no curvature, no offsets, etc) voltage over all PVT situations and a low noise content.
Also "low noise" and "accurate" are relative things. If "accurate" means +-5mV and "low noise" 1mV /rt-hz then all of this is meaning less. If accurate is 0.01mV and below, and noise measured in electrons/second, that's a different thing.
Don't get too excited, after all, its just electronics!