At first blush, the Mpemba effect does seem to be in play here. Unfortunately, the Mpemba effect is not very well specified and is generally studied over longer timescales than occur in the "boiling water to snow" phenomenon.
A quick quibble: the boiling water does not become snow, but a cloud of very small ice crystals, so it is more akin to cloud formation than snow formation.
The question is: Why does boiling water make an impressive cloud of ice crystals so rapidly instead of forming a layer of ice and then falling to the ground, and why doesn't cold water form the same impressive cloud?
Most explanations of the phenomenon allude to differences in surface area but neglect to mention how boiling water is able to increase its surface area more than cold water.
This site, while not a peer-reviewed publication, does a good job of discussing various possible contributions.
I have a few things to add:
In videos of the phenomenon, the cloud forms close to the surface of the liquid, leading to a "telescoping" effect as the water moves through the air.
One possible explanation of this is that the latent heat of freezing released by the water closest to the cold air heats the water further from the cold air. This layer of "inner water", since it is already close to boiling, will evaporate, dispersing the water molecules and quickly freezing, exposing and heating another layer, and so on.
It may seem strange that freezing water would heat surrounding water, but remember that the latent heat of freezing must go somewhere, and liquid water is more conductive than air. That, in addition to the rapid expansion and insulation provided by the water vapor (remember the Leidenfrost effect) is what allows it to disperse so quickly without bonding into a mass of ice.
I don't have enough of a background in atmospheric physics to point to relevant literature, but I hope this provides some insight into the subtleties of the phenomenon.