What is the purpose of the upper inductor in this speaker diagram?
But I couldn't figure out the the purpose of the inductor in series
  with the woofer.


It minimizes high frequencies into the woofer or, from the standpoint of overall load impedance presented by the filters and speakers, it's an attempt to maintain the impedance seen by the amplifier as roughly 4 ohms. As frequency rises the woofer circuit becomes progressively higher impedance and the tweeter circuit becomes progressively lower impedance.