Look, if you’re still having problems with the syntax for classes and methods, you really need to read the Java Tutorial or find some other source of information. All the information is there, and there’s got to be plenty of examples already posted on the Internet.
Swing is far more customizable than AWT. There may be a speed difference between the two, but both are pretty slow.
Try to avoid them both for the actual game in your program. If you’re making a Super Mario clone, there should be one panel (or whatever) containing the entire game area (i.e. where Mario can move around). You should store all the objects in the game yourself and draw them to this panel. Mario should not be a Swing component!!!
You can use components for things like the buttons on the title screen. They work great for that. There’s no way that your title screen is going to be too slow. You can even have buttons like that on your main screen, so long as you don’t do something silly like make your game characters components.
If you’re going to actually use AWT/Swing buttons, tables, etc., I would go with Swing. If you just need a frame to draw in, I would go with AWT because you shouldn’t need any of Swing’s features.