Some distillers estimate that the yeast contributes about 25 percent of the whiskey's flavor, so it's highly significant. What makes one yeast better than another I can't say. A distiller is looking for a yeast that is robust in its propagation action and, of course, one that makes the whiskey taste good.
There used to be a big distinction made between practical distillers and scientific distillers, with the former catching wild yeast from the environment and the latter using pure strain yeast, bred in a laboratory. Today that is somewhat blurred.
Yeast is a microorganism and there are an unlimited number of different strains. They are as unique as humans or, perhaps more accurately, human families. I'll resist drawing the analogy to the specific biological process, but the product of that process varies in subtle ways depending on the metabolism of the yeast family at work in a particular plant, hence the fermented mash and, ultimately, the whiskey has a distinctive taste.
To taste yeast characteristics you want to taste younger whiskeys. That is where the tastes from the yeast and grains are more pronounced, because there is less taste coming from the wood.