In dual federalism, both the state government and the federal government have their own jurisdiction and respect each other's jurisdictions in legislation without interfering. There is a clear difference in who deals with what and they don't intervene with one another when they are making policies. Because of this, it is also known as the layer cake federalism.
Cooperative federalism is the opposite of dual federalism. In cooperative, both the state government and the national government work together when making policies. It is called also the marble cake federalism because like the marble cake, it's all mixed and not clearly separated into layers.
An example of dual federalism in the United States can be when local, state governments, make policies regarding aid programs to people in the state. The federal government approves and allocates funds but the state can make policies regarding aid to some of its citizens independently of the federal government.
An example of cooperative federalism in the United States can be making policies regarding environmental protection. For example, the federal government can make a set of laws to protect the environment and reduce carbon dioxide emission, while a state government can create a local program that would help factories switch to clean energy in order to adhere to the law.
