Typically, firm strops are used pasted for abrasive stropping, while the hanging strop is used plain (w/o any honing compound) for finishing.
Both are to remove the root of the burr. The burr root is a scientific term, you may hear some call it the base of the burr. We take a blade to the strop having removed the bulk of the burr by other means of honing, AFTER we cannot palpate the burr (cannot feel it by brushing the finger across the edge).
Hanging strop is more pliant, "giving in", than a firm strop.
The firm leather strop is compressible only to a degree, and even in high-angle stropping can shape the root of the burr into the wire edge.
While the hanging strop "wraps" over the edge apex - this difference explains why the hanging strop removes the root of the burr better.
We define wire edge as a product of burr honing, when the burr root is shaped into an edge rather than deburred.
In our sharpening, knives of low HRC ductile knife steel often show residual burr even after high-angle honing on a rock-hard felt wheel, leather or paper wheel with fine diamonds - finishing these blades on the plain hanging strop helps to clean the apex of the burr root, and we can see instant improvement of sharpness on the tester.
Contrary to them, knife steels that are prone to form "negative burr" can be cleanly deburred with firm stropping at the edge angle or a little higher than edge angle, using the finest abrasive.
But I haven't done a comparison of plain firm to plain hanging stropping as this makes little sense in view of what we know about the burr formation and removal.