What are the advantages of using organic compounds to society, and what are the disadvantages of using them?

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.