How does the uneven heating of the earth result in the development of different climates?
Correlation between latitude and climate.
Because of the earth axis has an inclination the planet does not receive the same amount of solar radiation (heat) but instead such heating is dependent on the distance from the equator (latitude) and on the season.
As resulting effect some areas of the planet, the ones closer to the equator, are heated more and faster than the ones at higher latitude (closer to the poles).
In summer the northern hemisphere (north of the equator) receives more solar radiation (heat) than in winter making summer warmer than winter. The opposite happens in the southern hemisphere (south of the equator).
Because the earth spins the conductive air cells which develops as air is heated by the sun (similarly with the circulation in a boiling pot of water) are "twisted" by what is called "Coriolis effect" or "Coriolis force" generating specific regional and global wind circulations that affect the climate.
Finally the presence,or absence, of mountain ridges affects the local wind and humidity circulation and thus the climate. As example the Andes in South America and the Atlantis Mountains in North Africa block the humidity from the ocean to penetrate deep inland originating very arid climates in the areas behind the mountains.