How do people who live on the leeward side of a mountain climate differ from those who live on the windward side of the same mountain?
I am not sure what exactly you are asking here, as the people on the two sides of a mountain range are not necessarily going to be any different from one another. What I assume you are asking is how their weather is going to be different.
If that is what you are asking, the people on the leeward side of the range will see much less rainfall than those on the windward side. The reason for this is that the mountains will force the air upwards. This air will be relatively full of moisture. As it rises, the moisture will condense and fall. The air that crosses the mountain will be dry.
I have seen this effect most clearly when I lived in Hawaii. O'ahu, for example, has a windward side that is much more rainy and green than its windward side.
The difference on either side of one particular mountain may not be that great, and there are a number of variables that come into play, such as, is the mountain part of a mountain range or is it a stand alone peak? What is the elevation? Is it part of a ridge? etc.
I live on the eastern side of the Cascade Mountains in Washington State. people are usually surprised when I tell them I live in the desert portion of Washington, as they always picture rainy Seattle. This is called the "Rain Shadow" effect. In the windward side the weather systems move in and dump their precipitation as the elevation rises, then on the leeward side they proceed on, empty of moisture.