A meeting can be planned or unplanned. You can ring someone up and ask him to meet you somewhere at a particular time. It is also possible for you to meet someone quite accidentally. For example, you decide to go shopping. Your friend also decides to go shopping and you end up meeting him. In this case, the meeting is not planned; the two of you run into each other quite accidentally.
*We met in the library as planned. *The last time the two of us met was in London.
The word “visit” suggests a longer meeting. In other words, you are going to spend some time with the individual. You visit someone’s house. A visit is usually planned; the people you are going to meet usually know that you are coming.
*During the holidays, he visited his cousin in Pune. There, he met many of his old friends.