In the lab portions of the only two chemistry classes I took, we used deionized water, and even had our own little deionizers. Just because of this, I assume the deionized is the cleaner of the three. Note the use of the word "assume".
Not to contradict alien Traveler at all, but it's an experience I wanted to share.
This is some of what I fond on it: Deionized - Demineralized Water - Lenntech