Spring water is water that has overflowed from an underground water source. Distilled water has minerals and impurities mechanically removed from it before it's used.
Spring water bubbles up to the surface without having to be pumped. Some companies that sell spring water get it from springs that have been flowing for hundreds or even thousands of years. More conscientious companies protect their springs in order to uphold their claims of purity and naturalness. However, the law requires that the trucks that pump the water and carry it to the bottling plant be chlorinated in some jurisdictions. The water might also undergo some treatment once it gets to the bottling plant for decontamination.
Spring water that isn't filtered still isn't pure. Because of the way the water cycle works, the water is almost certainly full of minerals. Minerals add taste to the water and may even help the digestive system. For purer water, a person might need to turn to distillation.
With distillation, the water is heated until it steams. The steam is then cooled and condenses. The water that results is, ideally, pure water with neither contaminants nor health-giving minerals. Because distilled water lacks these minerals, some nutritionists believe it's not good to drink.