I can speak to the Second Bank of the United States (BUS). The Second BUS was created to regulate the economy and provide uniform economic stability akin to the Federal Reserve. However, it was privately owned with the government holding 1/5th of its stock. Of the 25 board of directors only 5 were appointed by the government not including the president of the BUS. With this sort of structure many felt that the BUS could easily succumb to corruption as the wealthy,most importantly foreign investors, could buy lump sums of stock and manipulate the economy. These worries of corruption proved true as during the Panic of 1819 a House Investigation of the BUS revealed a series of improprieties, fraud, and gross abuse of management. This sounded fierce movements to revoke the BUS's charter and reinforced debates of its constitutionality. The BUS was upheld as constitutional by the Marshall court in McCullough v Maryland and would remain in operation until Andrew Jackson took office when he abolished it and many accompanying forms of economic nationalism.

The First Bank of the US was created as a response to economic problems after the revolution, including United States war debt, inflation of the money issued by the Continental Congress, and individual bankruptcy and debt. After the War of 1812, war debt concerns and desires to have a uniform medium of exchange motivated the Second Bank of the US.
The Federal Reserve was established much later, in 1913, as a response to numerous bank panics caused by runs on banks, which occurred several times in decades prior, creating financial instability.

This time gap can partially explain why the current Central Bank is not called the Third National Bank--In the years after the Second Bank was dissolved, central banks in other countries were established with the purpose of serving as banker's banks, a term frequently used to describe the modern US Fed, and conducting monetary policy to maintain macroeconomic stability. The shift in the objectives for central banks probably seemed to merit a new name that confused the Fed less with previous national banks.
The differences in their origins and economic contexts suggest how they are different. The First Bank of the United States had operations that served mostly the government, including tax collection and some payment of US bills, which makes sense given the economic issues and financial needs that had to be managed by the new government. Most of this today is handled by other agencies. Both of the National Banks also gave out loans to individuals and businesses, which is something the modern Fed does not do.
An essential difference is that the National Banks did not do the business of regulating banks or the national economy through monetary policy. Although the Second Bank had as one of its programs a way of encouraging regional banks to not issue their own currencies, this is quite different from the responsibilities of regulation that the Fed has today.

I'm not certain how central banking today would be different. I'm mostly hoping I haven't said anything too wrong or misleading so far. My impression is that individuals and businesses have a "public option" for getting financing and could get loans at the Bank of the US, although the federal government does already offer many loans programs for specific things that it wants to incentivize, such as homeownership. It seems like the government of the US has evolved in such a way that would make a modern reproduction of the old National Banks largely redundant.

Sources:
The US Fed system tends to be pretty good about trying to teach the public, and often the websites of their branches are quite useful.
https://www.minneapolisfed.org/community_education/student/centralbankhistory/bank.cfm?&TC=1&DPR=1
http://philadelphiafed.org/publications/economic-education/first-bank.pdf
http://www.newyorkfed.org/aboutthefed/history_article.html