Rain isn't distilled water, as fumes from things like cars or other forms of pollution can cause the water to contain harmful components that can be removed using distillation (think acid rain). While the water cycle looks (at first glance) like a distillation process, it occurs in an environment that is quite unlike a laboratory, resulting in contamination.

As for electrolytes, usually you need to intake specific foods or liquids that are high in electrolytes (electrolytes are just chemicals that increase electrical conductivity of a solution when dissolved). So while a litre of tap water has more electrolytes than a litre of distilled water, a litre of fruit juice has more (though drinking a litre of fruit juice probably isn't great for you).

As for whether lack of electrolytes is harmful, the answer is yes. Electrolytes are required for your body's electro-chemistry to operate properly and maintain the proper function of your heart, brain, body's fluid-balance, and pH buffer system. When you lose electrolytes (such as through heavy sweating, diarrhea or vomiting) those processes may not work as well.

I would note that when you are dehydrated, hospitals will generally provide you with liquid electrolytes (intravenously if the situation is dire), implying that there is some medical benefit of liquid sources over solid ones (but I might be mistaken). But there are several solid sources of electrolytes.