Different types of sugars (sucrose vs fructose vs lactose) will likely crystallize differently (although I am not sure what sugar is best).

The type of sugar should not matter if it is fully dissolved, but if it is not heated high enough, it may very well affect the crystallization process.

Typically crystallizations work best with highly pure materials, so brown sugar likely wouldn't crystallize as well as highly refined (purified) white sugar. The crystal growth of a compound like sugar depends both on the different elements and the overall shape of a molecule. It is difficult to predict how differences will affect crystal growth.