Speaking with regards to my older 1D MkIIn, I love pressing focus and knowing that it's in focus. A DSLR's focus system calculates the difference, and tells the lens "move as fast as you can to this focus distance". A P&S generally racks the lens (pushes it back and forth) and watches for the image to come into focus.

The DSLR has bigger pixels. Within a given generation, the DSLR's noise will be orders of magnitude lower than the mini-sensor P&S. This is often the difference between getting the shot and not getting the shot. I love cranking the ISO and doing available light with my cheap 50mm f1.8 lens. This produces images that are incomparable to images taken with an on camera flash.

The larger sensor and dedicated lenses will always be able to have a narrower depth of field. This means that if I'm somewhere where the background detracts from a shot, I'll blur the background. I can also show the viewer of the image what to pay attention to, as the rest of it can easily blur away without even being apparent.

The interface is also designed for taking pictures. On my camera I can shoot a 6 hour event and never navigate a menu. I also can zoom to the point I want to zoom to in a fraction of a second. I can focus and recompose in that same fraction of a second.

However, all of these are nice to have, not required to take photographs. Every technique I use with my DSLR I can take and use with an old P&S, a tripod and a lightbulb. The images aren't as good, but they're certainly good enough to make a non pixel peeping audience happy.