No one framed drinking distilled water from time to time as being an immediately dangerous thing.
The effects occur over a long period of time for humans (much faster for smaller, more sensitive animals like fish in aquariums), and depends largely on your dietary intake - if you replace more salts with what you eat than you leach out with what you drink, you can drink all the distilled water you like.
That said, the effects do exist, and they can kill you - if you leach more salts than you replace.
Is tap water better for you than distilled water? A little bit, yeah - assuming we're discounting things like pathogens, which distilled water has far less of. Is it a big enough difference that you need to be worried? Probably not, unless you're not getting enough salts.
It's like the difference between eatting rice and eatting wheat. Rice technically is more calorie efficient than wheat, so it's technically better at preventing starvation. Except most of us don't live right on the edge of a healthy caloric intake, and changing from rice to wheat isn't going to cost us enough calories to make a difference and push us into the red unless we're already close to starving in the first place.
That said, you don't need to die to suffer adverse effects. If you're playing a competitive sport, for example, and you decide that instead of drinking the electrolyte-rich "Sportade" of your preferred brand, you'd rather drink distilled water, that can make the difference between having energy to keep playing, and feeling absolutely miserable and possibly even passing out from exhaustion and exertion.