This week's book giveaway is in the OCAJP 8 forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide and have Edward Finegan & Robert Liguori on-line! See this thread for details.
The main difference is that in Swing each component is painted(rendered), while in AWT "everything" is inherited from its native peer.
In particular, these AWT native peers allocate native resources. For example java.awt.TextField allocates an actual text-field-like resource* through the native OS while javax.swing.JTextField does not. JTextField simply draws pixels and interprets mouse events to emulate a native** text field.
*probably, depending on how AWT was ported to the native OS
**or other-than-native, depending on which LookAndFeel is in effect