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.
It makes no difference to the taste or texture that I've ever noticed. It does, however, slow the process down a bit as you have to bring the water back up to a simmer from cold.