design patterns – SimpleFactory vs Factory Method

Suppose a SimpleFactory that creates a group of objects:

public SimpleFactory {
   public Bycicle createBycicle(String type) {
     if(type.equals("ONE")) return new OneWheelBycicle();
     if(type.equals("TWO")) return new TwoWheelBycicle();
     if(type.equals("THREE")) return new ThreeWheelBycicle();

This has the advantage of centralizing the creation of Bycicles in a method / class. If new bicycles are added, there is only one place to change the code. So far so good.

What I don't understand are the benefits of the Factory Method. As far as I am concerned, I would always use SimpleFactory. Is there a factory method because a doSomething method includes code that manipulates the created object? Why does object creation include object manipulation? Or it may be because I can add n ConcreteFactories to group objects by factory, which I could also do in SimpleFactory by adding a new parameter for the group.

UML class diagram of the Factory Method