As you are probably already aware, distilled water and deionized water are NOT the same.
They differ in the manner in which they are produced, and if you are looking for one significant difference, it is that, in most cases, deionized water is more pure than distilled water.
Having said that, for most of the common reasons for using them, they are interchangeable.
Distilling water is fairly straight forward, you can even do it at home if you can be bothered. It is simply boiling the water, and then collecting and cooling the steam so that it returns to liquid state. Most of the ‘nasties’ in the water are left in the kettle, and the water that you end up with is far purer than it started off.
Deionising water is more complicated, and involves passing the water through a bed of Ion Exchange resin beads. There is a reaction that takes place which, in laymans terms, leaves the ‘bad stuff’ in the resin beads. For more information, you can find information, and even order, from this dionized and distilled water site.
Although there may be some instances where one is preferred over another, if you are simply wanting pure water for normal household applications, either will do just fine.