This week's book giveaway is in the OCPJP forum. We're giving away four copies of OCA/OCP Java SE 7 Programmer I & II Study Guide and have Kathy Sierra & Bert Bates on-line! See this thread for details.
I agree with Thomas. The Event model in Swing uses AWT. Also web application use applets which are easy to design and implement in AWT. (Swing based Web application require a component called HTMLConverter or Swing JRE on the client machine.
Learning AWT especially, the event model, it important for understanding Swing. Swing provides tremendous improvement over AWT as far as UI components like Table, TextArea, Tree, SplitPane, InternalFrame are considered.
I would for sure spend sometime understanding AWT before concentrating over Swing.