The bulletproof vest is made of kevlar fiber. The fibers are woven together. I wonder why we don't use kevlar as a sheet rather than woven fibers.
I rather suspect it's a matter of manufacturing ease. Here's some info describing two different bullet-resistant materials' production:
1 To make Kevlar, the polymer poly-para-phenylene terephthalamide must first be produced in the laboratory. This is done through a process known as polymerization, which involves combining molecules into long chains. The resultant crystalline liquid with polymers in the shape of rods is then extruded through a spinneret (a small metal plate full of tiny holes that looks like a shower head) to form Kevlar yarn. The Kevlar fiber then passes through a cooling bath to help it harden. After being sprayed with water, the synthetic fiber is wound onto rolls. The Kevlar manufacturer then typically sends the fiber to throwsters, who twist the yarn to make it suitable for weaving. To make Kevlar cloth, the yarns are woven in the simplest pattern, plain or tabby weave, which is merely the over and under pattern of threads that interlace alternatively.
2 Unlike Kevlar, the Spectra used in bulletproof vests is usually not woven. Instead, the strong polyethylene polymer filaments are spun into fibers that are then laid parallel to each other. Resin is used to coat the fibers, sealing them together to form a sheet of Spectra cloth. Two sheets of this cloth are then placed at right angles to one another and again bonded, forming a nonwoven fabric that is next sandwiched between two sheets of polyethylene film. The vest shape can then be cut from the material.