The History of US Paper Money. In the early days of the nation, ... The Massachusetts Bay Colony issued the first paper money in the colonies which would ...
www.ronscurrency.com/rhist.htm

That "feel of money" comes from at least three different things that make the paper in paper bills unique:

Normal paper that you use on a day-to-day basis (newspaper, notebook paper, paper in books, etc.) is made from the cellulose found in trees. Paper used for money, on the other hand, is made from cotton and linen fibers. This kind of paper is known as rag paper.
One big advantage of using rag paper is the fact that it does not disintegrate if you accidentally run paper money through a washing machine.

The paper used for money is thin compared to normal paper.

The paper used for money is squeezed with thousands of pounds of pressure during the printing process. This makes it even thinner and gives newly-made bills a special crispness.
The other special thing about the rag paper used in real money is that there are tiny blue and red fibers mixed into the paper when it is made. These fibers are easy to find in real money, but they are so fine that they do not reproduce very well in the counterfeit money from your inkjet printer.
money.howstuffworks.com/counterfeit4.htm

Magnet: The ink on US paper money contains a magnetic signature; a bill will be drawn toward an ... cotton and linen rag paper, used in US bills, will not. ...
www.pbs.org/wgbh/nova/teachers/ activities/2314_moolah.html

6) Get a high-quality rag paper that closely resembles the paper used in ...
used in money laundering. Exchanging bills for foreign currency and then ...
www.totse.com/en/bad_ideas/ scams_and_rip_offs/money-mk.html