I like both, but they do taste differently, and "ash" quality pretty much fits what I feel on the tongue. Also, maybe it's my imagination, but I think unpeeled ones are bit more acidic, I prefer unpeeled for red meat and dishes with strong taste, and peeled to go with chicken, sweet meats etc. But here your mileage may vary. Some people can't tell difference, some can always tell, and some - only sometimes.
According to these datasheets:
- Potatoes, boiled, cooked in skin, flesh, without salt
- Potatoes, boiled, cooked without skin, flesh, without salt
Per 100g serving potatoes boiled with skin provide 22% of vitamin c, and without - only 12%. Given it's water soluble, I think it's safe to assume that even if you discard peel, you still get more C if you boil with skin. Other differences are small. Largest are fluoride and ash. Even if the nutritional meaning of ash is a bit different, it really fits to what I taste. And vitamin C is acid - again, fits with how I can describe taste differences. Maybe that's just a coincidence, maybe not, can't really tell.