Vitamins are a totally different area, this is just about minerals, so basically salts of different types. Sodium, Potassium, Calcium and Magnesium salts are the most relevant ones as far as I know.
Any time you drink water that contains lower concentrations of these than your body, you will lose some, as the concentrations equalize. Same thing, just more pronounced. If you drink very large amounts of water in a short time, that can be a serious health hazard, it can lead to hyponatraemia [1]. Distilled water works exactly the same here, but it is more hypotonic, so the required amount to get into problematic areas is probably lower.
Short version: Don't drink huge amounts of water (think upwards of 5l) in a short amount of time without replenishing electrolytes, it could kill you. Applies to both "normal" and distilled water, but probably to a slightly lower amount of distilled water.
