Trace metals in water -- calcium and copper are examples -- can and will cause spots of DOS because these metals "catalyze" (accelerate) the breakdown of fats. Fat breakdown => Rancidity, aka DOS.
Hard water minerals in water -- calcium and magnesium -- will react with your soap and cause hard water scum (insoluble, sticky calcium and magnesium soaps). This happens in the shower, for sure, but it also happens in the soap pot. Soapers making liquid soap find this out the hard way when they use tap water to make their soap -- their diluted soap has an unattractive milky look. If they switch to distilled, their diluted soap is crystal clear.
I'm not saying DOS always happens to soap made with tap water ... but it can. And I'm not saying soap scum is all that bad ... but it can affect the skin feel of the soap. What I am saying is distilled water can help prevent problems that can be very mysterious issues to troubleshoot. Alternatives to distilled are reverse osmosis water or deionized water. If you have really soft water, that will prevent the hard water scum thing, although it won't necessarily prevent metal contamination.
Low tech alternatives for distilled water would be rain water (if you let the sediment settle out before use) or water from a humidifier (make sure the collection bottle is very clean). I wouldn't use either of these for lotion without thoroughly sanitizing the water first.
And, FWIW, tap, spring, and purified water are essentially the same thing.