Advantage of factory method pattern

From wiki,

The Factory Method design pattern solves problems like:

1. How can an object be created so that subclasses can redefine which class to instantiate?

2. How can a class defer instantiation to subclasses?

For example, MazeGame provides the instantiation capability to subclass like MagicMazeGame.

where,

public abstract class MazeGame {
    private final List<Room> rooms = new ArrayList<>();

    public MazeGame() {
        Room room1 = makeRoom();
        Room room2 = makeRoom();
        room1.connect(room2);
        rooms.add(room1);
        rooms.add(room2);
    }

    abstract protected Room makeRoom();
}

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Read answer.

Of course, this is a creational pattern, so solution should be around simplifying instantiation.

My question is,

What is the advantage of factory method pattern, that recommends a template method(public MazeGame(){..}) and defer instantiation to subclass MagicMazeGame or OrdinaryMazeGame?

Is it to abstract the intricacies of constructor details? in every class like MagicMazeGame or OrdinaryMazeGame