The Clone Stamp tool paints with a sample of an image, taken from a selection point. It can be used to touch up images. For example if you've got a scratched photo of a face you can use the clone tool to copy similarly coloured areas of flesh from somewhere else in that photo to hide the scratch.
The Pattern Stamp tool lets you paint with a pattern; the pattern can be an image you've created. For example, suppose you want tartan paint. You could create an image of tartan, set that as a pattern and and use the tool to paint objects.
Tutorials will show you when to use one rather than the other. Check the on-line help, your copy of Elements should include some basic tutorials.