See below for details
Hard paper was used by the National Bank Note Company and the Continental Bank Note Company. Soft paper was used by the American Bank Note Company.
The hard paper of the Bank Note issues is fairly white, perhaps it might better be called grayish white or sometimes a somewhat bluish white, while the soft paper seems slightly yellowish when compared with the hard paper.
Soft paper has a looser weave and more porous paper than hard paper, so it feels softer, displays a mesh or weave when viewed by holding the stamp between your eyes and light so that you are looking “through” the stamp.
Some people can also ID hard paper be “flicking” the edges and thereby “feeling” the stiffness of the paper versus the feel of soft paper if flicked in the same way. There's more of a snap to the hard paper.
On high magnification the perforation tips on soft paper will have more strands of paper sticking out than hard paper.
Soft paper is fairly dead looking under a long wave UV light ( (briefly and from a reasonable distance in a darkened room) while hard paper reflects more light. If reference copies of stamp designs known only on hard paper or soft paper are viewed under UV light, the difference in paper brightness should be apparent.
For a reference stamp obtain the inexpensive 1861 3¢ (#65), it is only available in hard paper.
A simple test is to hold a stamp to a lamp, you will see the hard paper is more translucent.