Weather

Weather was introduced to bring more variability into the game, and allow different strategies. Weather influences income of mana and gold, and also the price you have to pay when buying houses.
Different races prosper best under different weather conditions. Experienced players choose the best race for such weather that is in league they want to play. For example, races with more turns per day prosper better when houses are cheaper.

What is affected by weather

Gold income from population: from 50% (lowest) to 150% (highest income)
Mana income from magical units: from 50% (lowest) to 150% (highest income)
Price of houses: from 50% (cheapest) to 150% (most expensive)

Weather does not affect income of mana and gold from buildings!

Weather forecast

Weather forecast can change drastically from day to day (and so can the weather), so you should follow weather forecast and adjust your strategy accordingly. For example, invest more gold into houses than you normally would if forecast says that the price will rise next day. However, forecast is just what its name says - an estimate. You can't be sure what the weather tomorrow will really be. The only thing you can know for sure is, that if forecast says something "extreme" (drop or increase), it will never go in the opposite direction.