What are some examples of local government spending that is different than federal government spending ?

In a sense, local government spending and federal government spending are exactly the same thing.  The government hires people to perform services.  The government buys goods from companies.  In each case, the government has to spend money.  This is how government spending works regardless of what level of government is spending the money.

The difference between federal and local government spending has to do with what the money is spent on.  There are some things that are the province of the federal government while other things are up to local (or state) governments.  For example, a local government might spend money to maintain a local park.  The federal government does spend money on national parks, but not on local parks.  A local government might spend money on its police force.  The federal government has law enforcement (like the FBI) but it does not hire local police officers.  A local government might spend money on streets.  The federal government spends money on national highways/freeways, but it does not spend on local streets.  A local government might pay to put pipes under the ground to new developments.  The federal government does not do this.

All of these are examples of local government spending.  The spending itself is done just the same way federal government spending is done.  The only difference is in the exact things on which the money is spent.