Think of a singleton like a service. It's an object which provides a specific set of functionality. E.g.
ObjectFactory.getInstance().makeObject();
The object factory is an object which performs a specific service.
By contrast, a class full of static methods is a collection of actions that you might want to perform, organised in a related group (The class). E.g.
StringUtils.reverseString("Hello");
StringUtils.concat("Hello", "World");
The StringUtils example here is a collection of functionality that can be applied anywhere. The singleton factory object is a specific type of object with a clear responsibility that can be created and passed around where required.
