Conventional farming uses both pesticides and fertilizers. Organic farming may only use a small list of pesticides or fertilizers, which usually excludes most if not all conventional pesticides and fertilizers. However, this list is prescribed by the certifying body and may or may not admit synthetic pesticides.
Large-scale vs. small-scale farming
Conventional farming is usually geared towards scalable farming. Controlling pests without conventional pesticides is more difficult on a large scale. (A large organic crop would be the perfect breeding ground for pests.) So organically farmed produce is usually more likely to come from a smaller-scale farm.
Social/ethical aspects of paying workers
Many organic producers also support fair wages and are Fairtrade certified. However, there is no necessary connection between these two.
Quality is not clearly defined (taste, absence of imperfections, mineral/vitamin content, ...). In some cases, however, organically produced nuts have been shown to be higher in minerals and lower in metals, thus having a possibly lower harmful effect on the human body. See this study for more information.
Specific organic certification organisations
Standards for organic labelling are sometimes more, sometimes less strict. The same applies to nuts. There are also other labels that require a higher standard, such as demeter ("biodynamic" farming).
The final produce is the most interesting to the consumer, but only a small part of the growing process. Conventional pesticides often wear out the soil and present a danger to both wildlife (apart from the "pests") and the farmers, even if these dangers are tried to be avoided as much as possible.