The most important difference between compact P&S (Point & Shoot) and DSLR cameras is in the sensors. There is a huge size difference between the P&S sensor (often the size of a fingernail or smaller) and the DSLR sensor which can be the same size as a 35mm film frame (often called full-frame) or larger.
Regarding image quality, pixel size makes all the difference; the larger the pixel, the better its ability to capture light, and consequently, the less noise in the image. If you take a P&S and a DSLR sensor with the same number of pixels, i.e. 10mp, the DSLR will be capable of vastly superior image quality with less noise because those pixels will be much larger. This has nothing to do with camera brands or types.
As a commercial/advertising shooter for over 40 years, I started using digital in its early years. Today's sensor technology is amazing compared to the early versions, but you can't change the laws of physics, and pixel for pixel, larger ones still win hands down! These days I shoot digital exclusively, using numerous camera bodies from different makers. My favorite camera has interchangeable lenses but is neither a DSLR or P&S.
Another factor to consider is the image color quality of the the sensor. Here again, most DSLR sensors have an edge. Since each sensor photosite (that is what they are called before they become pixels) is covered by a red, green, or blue color filter, the quality of those filters also affects the final image quality. Once the raw capture data is converted into an image file, the photosites officially become pixels.
There are many pitfalls in the path to a final image and one of them is the conversion from captured data to an image file. Many DSLR cameras allow you to save the captured data as Raw files for later conversion on your computer as JPEG or TIFF files, or converted in the camera to JPEG files. Only a few P&S cameras allow saving in the Raw format.
A discussion about Raw vs. JPEG files is outside the scope of this thread, but without a doubt, a Raw conversion made on your computer has far more potential to make the best quality prints. I know this is an overload of info, and I am only scratching the surface, but this discussion could easily go for several pages.
One final note - if you are only interested in 3"x5" machine prints from a local drugstore or Costco, then all of this discussion is academic. On the other hand, if you make prints up to A1 size (23"x33"), then all of this is a valid concern.