Everybody know that search engine like Google hates duplicate content. You may think you write your own content and published no where else. Thus, you have no duplicate content at all. Unfortunately, http://yoursite.com and http://www.yoursite.com are not same website in search engines' eyes. That's also known as canonical problems among webmasters. Even though you don't need to worry too much for insite duplicate content, It's better if you have no duplicate content at all. So what you should do is that permanently redirect (301 Redirect) non www url to www url or do vice versa. In that way, the visitor is automatically taken to the canonical url even if they typed the wrong url or clicked the wrong link.
After the old web page has been moved, you should make a 301 redirect to allow visitors or search engines to access it. There are many way to do that. This is one of them.