Interchangeable lenses - so a big choice of focal lengths and the opportunity to keep buying new higher quality ones as you can afford them.
Larger sensor, so less image noise (I think you'll be genuinely surprised if you open up a 3MP P&S digicam image to 100% versus one from the 3MP D30). A lot of people think that megapixels = quality, and while for digicams for general use, it's often a good indicator, it's not the whole story. The type and size of sensor and processing chip play a big part as well as how the pixels are arranged on the sensor
There's often a lot more functionality on a DSLR - such as being able to shoot in a continuous focusing mode (AI Servo on Canon DSLRs) which will keep refocusing while you have the shutter button half pressed, or being able to select your metering mode. These are justa few examples (there are more) and while they aren't specific to quality, they all combine to make a better photo - eg some digicam users will complain of shutter lag when taking action photos. On a DSLR in AI
Servo, you are following the moving subject and refocusing all the time and can just let the shutter go at the right moment and you have a very very small lag time. Can mean the difference between an out of focus shot and a crisp sharp one.
You will also be able to shoot in RAW format, which gives you more control of your images after you have taken them. It can sometimes mean the difference between salvaging a photo that wasn't taken correctly. In RAW, the camera records the settings that you had on the camera, but doesn't apply them to the image and compress and save as JPEG. You can go in afterwards and change settings and then convert the photo to TIFF or JPEG to edit in Photoshop.
Not to take anything away from some high end prosumer digicams, which often have lots of the same features of DSLRs, but with a fixed lens.
Hope that helps a bit - I'm no expert....though I'm sure a few might come along and reply!