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Point and shoots usually have pretty bad shutter lag, so you tend to miss shots because they've moved after pressing the button. The lens quality isn't as good as the more expensive lenses that come with a DSLR. They are going to be much slower, so you'll need more light to take hand held shots. You'll probably be able to see lens distortions at more of the zoom range than with a DSLR lens. The sensor in the camera is going to be noisier than in a DSLR at the same ISO.
Generally, PASs are for people who want to take snap shots for their web page and aren't very serious about photography. The cameras that have been talked about in this thread are pro-sumer and are for when people start taking it more seriously.
Can you take stunning pictures with a PAS? Yup. Just not as easily and not in as wide a range of lighting situations as with a DSLR.
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Good response. (except the slower speed comment P&S lenses are usually fairly fast. OP's S3 opens up to f/2.7. You generally can't get f/2.8 in a DSLR zoom lens without spending $900+).
The main advantages of a DSLR over a P&S:
1) focus speed/shutter lag. These are interchangeable because one of the reasons for shutter lag in s P&S is because it takes longer to focus. So, if you're taking photos of a moving target (e.g., your kids), you're a lot more likely to get an in focus shot using a DSLR.
2) Lighting flexibility. Most P&S cameras have a built-in flash with no option to use an external flash. The problem with the built-in flash is that it is too close to the lens (leading to red-eye), and it is not powerful enough (leading to dark photos and/or dark backgrounds. An external flash is much more powerful and is farther from the lens. Plus, you have the ability to move the flash off the camera. (You rarely see "professional" portraits taken with a flash located on the camera.)
3) Better high ISO performance. Today's DSLRs are very good even at ISO 800 or ISO 1600. Many P&S cameras are practically useless above ISO 400 because of noise. (compare
http://www.dpreview.com/reviews/CanonS3IS/page6.asp with
http://www.dpreview.com/reviews/canoneos40d/page19.asp and
http://www.dpreview.com/reviews/canoneos400d/page18.asp
Those are the main advantages, but as one becomes more advanced, they can branch out into different fields of photography, each of which has specialized lenses that are much better than P&S lenses (e.g., portrait lenses, long lenses for birding/sports, macro lenses for close-ups).