What are the advantages of using organic compounds to society, and what are the disadvantages of using them?
According to modern definitions, organic compounds contain both carbon and hydrogen atoms. These compounds are part of living things or produced by living things. Important organic compounds include carbohydrates, proteins, lipids. Carbohydrates are made of the monosaccharide glucose, which is used in cellular respiration to yield A.T.P., a high energy molecule that gives cells energy. Proteins are important organic compounds which provide amino acids, essential to make proteins in the body. Lipids contain great amounts of stored energy that we can use in our own body as well as for making products including cooking oil, candle wax, lubricants, etc. These are all advantages to using organic compounds. Fossil fuels are hydrocarbon fuels, derived from plant remains or the remains of microscopic organisms. They contain vast amounts of stored energy. However, when burning coal, oil and gas for energy, greenhouse gases are released. These include carbon dioxide, carbon monoxide. When these are released into the atmosphere, they absorb heat energy from the sun, acting like a greenhouse, causing global warming. This is definitely a disadvantage to the use of organic compounds, as an energy source, however, the advantage is that these compounds are currently our most inexpensive way to get the energy we need to provide fuel to our vehicles and to generate electricity.
It sounds like you might mean simply "organic", rather than "organic compounds". trophyhunter answered your question as written--organic compounds are indeed those that contain carbon, and Organic Chemistry is an entire course on just those compounds. The other use of the term "organic" is supposed to mean, generally, earth-friendly, or not using artificial chemicals in food, agriculture, or manufacturing in general. The legal and general use of the term are different, and has evolved as food producers have used misleading advertising to make their products seem healthier than they are. It is only in the last hundred years or so that this has even been necessary--before artificial additives and fertilizers, everything was already something that was found in nature.
Benefits are that when we eat fewer additives, we may decrease our risk of diseases such as cancer; we don't really know what long-term use of some of these chemicals can do, both to individuals, humans, and the earth overall. Problems arise when people don't really understand what the term means, and reject everything that is not "natural"; getting diseases is "natural", for example, but we don't want to go back to getting smallpox!
I've attached a couple of links--this is a complicated subject, with long-term effects we don't really know, yet.