They will need the plug-in to use Swing in a browser, as the browsers only support java 1.1.something that does not support swing.
There are many benefits. First of all, there aren't peer components in swing. For every AWT component, there is a native peer component in the OS. This slows things down, and is restrictive. Because of this, there is also the pluggable look and feel, which allows you to change how everything looks. You can make your whole program look like windows, unix, or java's own "metal" look, and it will look that way on all platforms. You can also create your own. Swing also has more components than AWT does, and most things are more customizable.
You can find some more information here: http://java.sun.com/products/jfc/