First, there are plenty of stone buildings in Asia. In particular, brick and stone are the main building material in much of Western Asia through to northern India, through Central Asia into Tibet, western and northern China, and Manchuria. The "Asian wood-building zone" is largely non-arid India, South-East Asia, southern and eastern China, and Korea and Japan.
Second, the predominance of stone and brick correlates with the scarcity of wood. The three most common building materials are wood, stone, and brick. We can also add rammed earth as a common 4th material. Fired brick and stone are functionally similar, and unfired brick and rammed earth are functionally similar. Given a wooden frame, a variety of materials can be used for non-load-bearing walls; some example include brick, stone, wood, wattle-and-daub, felt, large leaves. Roof materials favour water-resistance and light weight, but we still see stone (tiles) and fired clay (tiles) in use, along with wood (e.g., shingles), thatch, and more.
Traditional housing is usually built with locally-available materials. Why? Cost! Before modern transport, stone and brick were expensive to move long distances, so stone was usually quarried locally, and bricks made locally. Trees for wooden construction were usually felled locally. So availability affected the choice of materials. Still, often, multiple materials were available. So why choose one over another?
Stone is expensive to work, but fire-proof and weather proof. Thin stone walls are not very strong, so they tend to be made quite thick, which adds to the cost. If building a fortress that needs to resist artillery (gunpowder or mechanical), one wants thick walls anyway, so this isn't a further burden. If one wants large airy windows and doors, this is difficult to achieve with stone (but a combination of stone and wood can be very successful).
Fired brick: largely similar to stone, but often cheaper, and standard-sized bricks can make it easier to build thinner walls. However, thin brick walls are very susceptible to earthquakes (also a problem with stone walls).
Unfired brick and rammed earth: cheaper than stone and fired brick, but much less weather resistant, and not as strong. Walls need to be thick. As with stone, large airy windows and doors are difficult.
Wood is relatively light and strong, and can be easily used for spanning large gaps, allowing large open spaces (thus, wooden roof framing is often used with brick/stone walls). It is flexible, and earthquake resistant. The disadvantages of wood are that it can rot, be attacked by termites etc., and burn. The first two of these limit the life of wooden buildings, unless they are regularly maintained (including the replacement of timbers when necessary), and make wood unsuitable for foundations. The last of these is not as bad a problem as it is sometimes made out to be - large framing timbers do not catch fire easily, and even when the building burns, can survive the fire. What is quite bad for fires is more flammable walling between the main timber frames, and roofing. While stone/brick has a reputation for fire-resistance, a stone building with a thatched roof is quite susceptible to fire. East Asian construction with a stone foundation, wooden walls and roof framing, and a tile roof at least provides a highly fire resistant covering on the roof, reducing the chance of fires spreading.
So, parts of Europe which are tree-poor often have a long tradition of stone building. Wet tree-poor areas favour stone over brick (because unfired brick and rammed earth are eroded by water, and fired brick is wood-hungry (or, later, coal-hungry)). Note that in England, the use of stone and brick has increased over time, matching a decrease in available wood.
Dry tree-poor areas favour unfired brick and rammed earth over stone on the basis of cost. Thus, the common use of these in arid areas, where the effect of rain on unfired brick or rammed earth is a lesser problem.
Where wood is plentiful, wood is often the preferred building material. Due to the ease with which wood can span large spaces, wood roof framing is often used with stone and brick walls. Where wood is the primary building material, it is often combined with a brick or stone foundation. In both of these cases, the roof is wood-framed, and the foundation is stone/brick. The difference is in the material of the load-bearing part of the walls and the material of the rest of the wall.
References:
For a nice collection of photos of traditional Chinese buildings, see Peking University Library, A Pictorial Record of the Qing Dynasty: Qing Dynasty Architecture, Cengage Learning Asia, 2010.
Chapter 14 in Anthony Reid, A History of Southeast Asia: Critical Crossroads, Wiley, 2015 discusses the choice of building materials and changes due to deforestation and regulations to improve fire resistance, earthquake resistance, and rat resistance.
For more on Asian architecture and it's history, see Vimalin Rujivacharakul et al., Architecturalized Asia: Mapping a Continent through History, Hong Kong University Press, 2013.
