How do polymer properties change when moving from bulk to nanosized fibers?
Nanofiber polymers in contrast with bulk polymers have high ratio of surface area to volume.
So, nanofiber polymers could be useful for any application that needs to maximize the ratio of surface area to volume.

Let's consider their application as scaffolds for tissue engineering: increasing the surface area of artificial-organ scaffolds gives cells more points at which to adhere.

Another application of nanofiber polymers is their use as electrodes  in fuel cells:  the greater the electrodes surface area, the more efficiently they catalyze the reactions that drive the cell.

Another interersting application of polymer nanofibers is in meshes that allow only nanoscale particles to pass through: ultrafine air, water filters, membranes for fuel cells.

Finally, as the capacitance is proportional to the surface area,  capacitors that store electricity, are one example of the possible applications of  polymer nanofibers.

For more details, please see this site and this article.