Most artificial flavorings are derived from petroleum. They may affect RNA, thyroid, and enzymes. Most have not been studied for safety or toxicity. They are all synthesized chemicals that don't even have common names. Most artificial flavors actually contain many chemical ingredients, not just one. Many of those chemicals are volatile. Natural flavors are chemicals, too, though.
The FDA doesn't require manufacturers to list color or flavor additives on ingredients left, as long as they are recognized as safe. Some of these can cause allergic reactions in intolerant people, and these folks can find it hard to avoid those substances. Many times the ingredients will simply say "artificial flavors" without mentioning which ones. Most of them don't have common names anyway.
Companies have been trying to use natural flavors in their products for the last 20 years. Many of the synthesized flavors can actually be found in nature. Processors have been trying to derive these flavors from nature rather than synthesize them. It is better to get these substances from nature than from petroleum, obviously, but synthesizing them can allow us to create purer versions that are less dangerous.
Some artificial flavors are safer than their natural counterparts. Benzaldehyde is almond flavor, and when derived from nature, traces of hydrogen cyanide--a deadly poison--can be found in it. When made by mixing oil of clove and amyl acetate, no cyanide is produced. Artificial flavors are actually better in many cases, because they don't contain contaminants or toxins. However, benzaldehyde made either way can cause central nervous system depression and convulsions.