The large banknotes are those stamps from Scott #134-218. Typical designs are shown below from my inventory.
Scott 137 used pair with New York Foreign Mail Geometric
Scott 210 Previously Hinged Graded 95
The large banknotes were printed by three different printing companies. These printings are as follows.
1870-71 Scott numbers 134-144, the grilled issues. Printed by the National Bank Note company on hard white paper.
1870-71 Scott numbers 145-155 no grill. Printed by the National Bank Note company on hard paper.
1873-1875 Scott numbers 156-181 including special printings. Printed by the Continental Bank Note company on hard paper.
1879-1890 Scott numbers 182-218. Printed by the American Bank Note company on soft porous paper.
Many of these design are the same. The appearance of Scott #150 and 187, the 10c brown Hamilton, is virtually identical except for the paper types, 150 being on hard paper and 187 on soft paper.
The best way to distinguish the paper types is to hold the stamp up to a strong light. Look along the edges of the stamp between the design and perforations where the paper is white. The hard paper will be clear and translucent. The soft paper will be mottled in appearance, showing the paper weave. To practice use a common Scott 65, which is always on hard paper, and a Scott 210 or a low value Columbian such as 231, which is always on soft paper. Hold the two stamps up the the light and you will clearly see the difference.
Important note. Hard and soft paper varieties also occur on newspapers and officials. Starting in 1879 at Scott number O94, the Officials were printed on soft paper. Similarly in 1879 starting with Scott number PR57, newspapers were printed on soft paper.
There are examples of intermediate papers that are hard to distinguish but for the most part, with a little practice, looking through the papers will allow you to tell hard from soft paper.
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