Washi
Why the paper matters.
Washi is made from long plant fibers rather than wood pulp. It folds without cracking along the crease, holds a sharp wing, and does not go brittle in the months between an order arriving and the wedding day.
That is the practical reason. The other reason is how it takes light — slightly uneven, never flat, which is what makes a table of one hundred cranes look handmade rather than printed.