Why does sodium polyacrylate solidify, on addition of water?
Sodium polyacrylate is an example of a super-absorbing polymer. It is a cross-linked (network) polymer that contains sodium atoms. It absorbs water by a process called osmosis. When the (sodium-containing) polymer is placed in contact with water, there is a tendency for the sodium to distribute equally between the network and the water.

That means, some of the sodium atoms want to leave the network and move to the water. When these sodium atoms leave, they are replaced with water molecules. Water swells the polymer network to try to keep the sodium concentration balanced between the polymer and the water.

The cross-links that connect the chains together prevent them from dissolving/breaking apart in the water. The resulting product appears very similar to fake snow, as you mentioned, a solidified powder like substance. 

Sodium polyacrylate can absorb it’s own weight 300x over making it an amazing chemical to aid in water cleanups.

The question was good.