The most widely accepted definition of Organic wine is wine made from grapes grown in accordance with principles of organic farming, which typically excludes the use of artificial chemical fertilizers, pesticides, fungicides and herbicides. The legal definition of Organic Wine is a complex issue and varies from country to country. The primary difference in the way that organic wine is defined relates to the use (or non use) of preservatives during the wine-making process.
In the U.S., the creation by the USDA of a NOP (National Organic Program), an organic wine is defined as "a wine made from organically grown grapes without any added sulfites". This means the vast majority of what you and I may have been calling organic wines can now only be referred to as "wines made from organic grapes" (or organically grown grapes), since they are allowed to contain up to 100 ppm of added sulfites.
Organic wine refers to the absence of chemicals in the growth of grapes and in the winemaking facility and production process. It is a legal certification with third party inspection.
Natural winemaking is a style of winemaking that can be applied to any wine. It is loosely defined as using native yeasts in the fermentation process and minimal or no sulfur dioxide in the winemaking process. It may also mean unfined and unfiltered as well. Natural winemaking is not governed by laws (at least not in the U.S.) and has no inspection or verification process (unless it is a biodynamic wine).
Some farmers take additional steps beyond standard organic winemaking to apply sustainable farming practices. Examples include the use of composting and the cultivation of plants that attract insects that are beneficial to the health of the vines. Sustainable practices in these vineyards also extend to actions that have seemingly little or nothing to do with the production of grapes such as providing areas for wildlife to prevent animals from eating the grapes and allowing weeds and wildflowers to grow between the vines. Sustainable farmers may use bio-diesel for tractors in the vineyards to reduce emissions among the vines, or plough with horses.