What expenditures do state and local governments have that the federal government does not? Why?
There are not many areas in which state and local governments spend money and the federal government does not spend any at all. This is because the federal government has such a big role to play in our lives.
For example, state and local governments pay for policing and fire fighting. But the federal government has the FBI (a police force of sorts) and gives grants for various policing programs to the state and local governments. The federal government also has firefighters who fight fires in national forests and other such places.
As another example, the states and localities are largely responsible for funding education but the federal government also gives grants to help with this. Federal monies account for something like 10% of education spending. It is this money that allows the federal government to push the states to abide by laws such as No Child Left Behind.
A third example would be parks and recreation. Obviously, the federal government does not pay for local parks. But it does run the national park system.
There may be a few areas in which the federal government is not involved at all. However, because of the size of the federal government there are few, if any, such areas.