Tap or well water that is perfectly safe to drink can contain calcium and magnesium and other chemical impurities. These chemicals can affect the quality of your soap in ways that might not be ideal. Metallic contamination can trigger DOS (rancidity). Calcium and magnesium will react with the fats to make their own insoluble soaps -- aka "hard water scum". The impurities in tap water can vary with the season, the source of the water (for example, big cities often use water from rivers supplemented by water from wells), the type of pipes used to deliver the water, and the type of water purification treatment.
There's no rule that one MUST use distilled ... if tap water works for you, it works. But if you have odd problems pop up, don't rule out that your tap water might be a culprit. I prefer distilled because I prefer to do my chemistry as an engineer or chemist would do it, since that is my professional background. If I didn't have access to distilled, I'd use filtered rainwater instead.
As long as distilled water kept in a chemically-inert, tightly closed container in a clean environment, it will be fine -- your goal is to simply maintain the chemical purity of the water. There are two reasons why distilled water doesn't taste especially good. One is there are no dissolved minerals to give the water a "good" taste. The other is there is no dissolved oxygen, which also enhances the taste of water.