While not a commonly asked question, this topic has come up in the past so we feel as though it deserves a moment in the water quality spotlight. A quick search on Google turned up the following piece of information which, we think, addresses the matter quite nicely:
“Deionized water is water that has been passed through a column or membrane to remove ions present. If it is of the type used in homes, it is not truly a de-ionizer, removing all ions, but rather an ion exchange column that exchanges polyvalent ions such as Mg++ and Ca++ for Na+ ions. A de-ionizing column will not remove nonionic organic substances from the water.
In contrast, distilled water is actually boiled in a still and the condensate collected and distributed. Distillation removes both ionic and nonionic organic contaminants.” (source)
For the layperson, that last quote basically says properly distilled water will contain neither ionic contaminants (such as metals) nor nonionic contaminants (such as bacteria) while properly deionized water may still contain nonionic contaminants (such as bacteria).