As far as I can tell, it doesn't really do anything, except be more technically accurate when talking about a climate (which really means "temperature" in-game right now). However, biomes are more relevant to the game, and if we already have temperature and rainfall meters, the only thing that might possibly be helped by this is the notion of how seasonal the rainy season is... and we could probably do that by just having another variable for seasonalness of the rain.
Essentially, "Climate" just doesn't fit into this game.
We don't need another organizing classification system into a game where we already measure all the constituent parts of what goes into climate, and already have a much more specific biome system to do the organizing. We already have a measurable rainfall, we already have a measurable temperature, we already have salinity, we already have distance from oceans and rainshadows from mountains.
Having a more distinct mechanic where the closer you are to an ocean, the more stabilizing the effect upon the climate becomes (especially if DF were to model oceanic currents, so that, for example, ocean currents form clockwise in "northern hemisphere" regions, carrying warm water up east-facing coasts, and cold water down west-facing coasts), and the seasonality of rain (which may simply be related to being near a equator-facing oceanic coast in more tropical temperatures) would naturally be an improvement, if a minor one that only appears for those who truly look for it (or if certain crops are monsoon-dependent). This wouldn't need to include "climates", however, merely more specific world generation mechanics and weather controls.
Simply let this form into making the notion of the biomes, and it will be a more technically accurate system than a "climate" system, which is really just an organizational construct, not an actual phenomenon.