The difference between distilled and purified water lies in the process that the water goes through; distilled water goes through one specific process, while purified water can go through a number of different cleaning processes and still have "purified" on the label. The distillation process boils all the contaminants out of water, but purification takes on a number of different guises.
Boiling the contaminants out of water requires elevating the temperature beyond water's boiling point and capturing the steam that forms. As that cools, it becomes the distilled water, and the leftover elements are the contaminants. One problem with distillation is that pesticides and other toxins have lower boiling points than water, meaning that those elements end up in the steam as well. More purification steps are necessary after distillation to get perfectly clean water.
"Purified water" has a set level of impurities that must be reached to carry the term on the label. The level of dissolved solids has to be less than 10 parts per million, and water at this point is purer than tap water, spring water or filtered water. Filtration is less rigorous than purification, so reading "filtered" on the label does not mean that the water is pure.