It adds emphasis. Depending on the context, it might be emphasising how quick something happened. e.g. I told him about the new competition, and he put in his entry the very next day. Was it the same dress she was wearing? Yes, the very same one.

It's just stressing or emphasising that it happened the next day. You add the "very" to emphasize or stress the word that's following. I was sick versus I was very sick. I was tired, I was very tired. Here you are emphasizing that you were not just tired or sick, but you were "really" sick and "really" tired. Similarly, when you say you got a call the VERY next day you are stressing it was the next day. Not sooner, not later.

No difference. Another example: He did the same thing the next time. He did the same thing the very next time. Still no difference.