Comparing two classes by its types or class names

There is need to compare two objects based on class they implement? When to compare using getClass() and when getClass().getName()? Is there any difference between this approaches to compare two Objects class types (names)?

public abstract class Monster { ... }
public class MonsterTypeOne extends Monster { ... }
public class MonsterTypeTwo extends  Monster { ... }

    Monster monster = MonsterTypeOne();
    Monster nextMonster = MonsterTypeTwo();


if(nextMonster.getClass().getName().equals(monster.getClass().getName()) )// #1

if(nextMonster.getClass().equals(monster.getClass()) )// #2

EDIT 1

What about: ?

nextMonster.getClass().equals(MonsterTypeOne.class)