Despite what you might think, there's more than one way to look at sneezing and coughing. They can be symptoms, in which case your first reaction is to try to "cure" them with all sorts of pills and syrups. Or, you can see them as the body's natural defense system. It's called Darwinian medicine.
Darwinian medicine tries to find evolutionary explanations for why we get sick. It also tries to figure out how and why we respond to sickness. For example, when you get a low grade fever it might be because your body is trying to make things uncomfortable for invading microbes. When you take medicine to lower a fever you might actually be helping whatever's making you sick in the first place.
Sometimes medicine is absolutely necessary, like when you have a really high fever. But the point is that it's not always best to automatically try to stop all symptoms when you don't feel well. Until studies are done on the effectiveness of taking aspirin and other fever-lowering drugs, it may be hard to know when to treat cold symptoms and when to leave them alone.
But in the case of a cold or mild fever, it may very well be worth letting nature take care of business. Bear in mind, too, that aspirin should never be given to children.