The two can be quite similar, but remember that the true Singleton must itself be instantiated (granted, once) and then served. A PHP database class that returns an instance of mysqli is not really a Singleton (as some people call it), because it is returning an instance of another class, not an instance of the class that has the instance as a static member.
So, if you're writing a new class that you plan to allow only one instance of in your code, you might as well write it as a Singleton. Think of it as writing a plain-jane class and adding to it to facilitate the single-instantiation requirement. If you're using someone else's class that you can't modify (like mysqli), you should be using a static class (even if you fail to prefix its definition with the keyword).
