What do you mean you can't use "Nombre"? There is no difference, only that "Mi nombre es" = "My name is" "Me llamo" = "I am called" In English nobody says "I am called", but in Spanish it's very common, you can say either when you introduce yourself, you can also say "Soy Carlo" = "I'm Carlo" (which is also common in English).

Me llamo is more common. This is partly because "nombre" really means "first name" so it sounds a bit odd to say something that is the equivalent of "my first name is _____ "