Does it change the taste, the texture, or anything else?
And no, I am not referring to potatoes or anything specific. I am talking about vegetables in general.
This is my personal experience. It takes sometime to reach the boiling temperature (again) when you add colder water. As a result, vegetables get cooked for a longer period than anticipated. This makes some vegetables becoming mushy and not tasting as good as otherwise. (Example: Egg Plant) Therefore, I make sure to add the right amount of water in the first place.