Here's a fourth--if the carbon dioxide we exhaled didn't dissipate quickly, we would all be at a constant risk of suffocating ourselves. Instead of free oxygen being readily available to breathe in, we would all be sitting around in our own personal clouds of exhaled carbon dioxide! This kind of ties in to the above editor's answer of the oxygen from photosynthesis needing to diffuse away from the plants where it is being produced, and in fact, our exhaled carbon dioxide also needs to make its way over to the plants that need it.
To add to his answer about the air pollution, in that case it is lucky that the hotter gasses tend to rise. It gives more area for the gasses to mix with others as the volumn is increased.
Diffusion is the process by which molecules of a gas move from a more concentrated state to a less concentrated state because they are constantly in motion. All gases are constantly in motion and this is the basis of the kinetic molecular theory. You may experience this when someone standing near you uses perfume or a spray cologne and you quickly detect the odor of the perfume.
Three examples that come to mind are first the detection of leaks of natural gas when you smell the odorant that is added to natural gas. Natural gas has no odor of its own and so a strongly smelling sulfur compound is added. This way if a leak occurs it can be quickly detected and fixed, minimizing the possibility of an explosion.
A second example would be the release of oxygen from plants during photosynthesis. This oxygen is necessary for all animal life on earth and if it did not quickly diffuse some areas of the world would have high levels of oxygen while other areas (such as deserts) would have almost none.
A third example is the exhaust from automobiles. This exhaust contains, along with carbon dioxide and water, many other compounds, some of which are harmful and even potentially fatal, to people. Because of diffusion, the gases in the exhaust quickly mix with the surrounding air and drop to levels that are no longer dangerous to us.