Many people have asked this question but it is not easy to answer without knowing what you use the camera for. I have both kinds and I can share my experience.
First of all, I agree with everything said above. It is absolutely true that a nice bulky camera that stays home will not do you any good for some spontaneous moments of photo opportunities. It is also absolutely true that your creativity and photography lies in you and not the camera. There are professional photographers that use advanced compact instead of D-SLR for their projects.
The PS camera is compact and light, and can travel with you anywhere. It is generally easier to use if you are the kind who like automatic operation. Most PS cameras nowadays can take good to excellent photos. One advantage (or disadvantage) of PS camera is the greater depth of field than D-SLR. A greater depth of field will keep everything better focused in the photo including the wanted/unwanted background. A shallower depth of field when using a large aperture lens with D-SLR will keep the subject in focus but the background blurred (this is done often with portraits). So if you are taking tourist type of photos with your family standing in front of the beautiful scenery, the PS camera will keep the background scenery in better focus than the D-SLR in general. The D-SLR may have problem blurring the subject or the background scenery. Of course the main disadvantage of PS camera is the lack of flexibility and customization. It is harder to use the manual functions with cumbersome hidden menus. It also has poorer low light high ISO performance, and has limited capabilities for low light photos, and more than frustrating performance with low light action. You can of course try to overcome the low light problem by using flash and tripods. But low light action remains a challenge to PS cameras.
The D-SLR camera is actually quite easy to use, with more options to change lenses, customize settings, and room to expand your creativity. It has faster and better low light autofocus and high ISO performance, and easily outperforms PS camera in low light actions. The main disadvantage is its bulk and cost. You often have to carry the camera body and at least 2 lenses to be equivalent to one PS camera. A good general purpose lens for D-SLR usually is no more than 3-4x zoom equivalent of a PS camera. There are ultrazoom lens that has more than 11x zoom equivalent, but these usually have lower quality optics and may not be that much better than PS cameras.
You probably know all the above already and can read widely anywhere on the web. But let me share with you about my personal experience and you can see whether this answers your question. The main reason I buy a D-SLR is because it outperforms PS camera in low light action, so I can capture my fast moving 3 year old in any situation. To really be able to achieve this and notice a big difference from my PS camera, I need to buy a large aperture zoom lens (or you can buy large aperture prime lens) with at least f/2.8. These lenses cost more than the D-SLR. Yes, the D40 with kit lens are quite cheap, but it cannot take low light action photo without flash or tripod. Using flash is too slow since it will slow down the shot-to-shot time significantly and I will miss the moment, not much better than the PS camera. A Canon or Nikon general purpose zoom lens with f/2.8 aperture with image stabilization can do the job, but will cost about $1000. And a tele zoom lens with f/2.8 and image stabilization costs close to $2000. And you will need both lenses, which will put the cost up to $3000 without the camera body. Getting a cheap slow kit zoom lens will not give you good enough low light action performance. Of course the D-SLR with kit lens can freeze action in low light with flash, but so can PS cameras (though D-SLR will still have faster autofocus and shot-to-shot time than PS cameras).
If you do a lot of portraits and like to have blurred backgrounds, then D-SLR will shine over PS cameras too, but again you will need a large aperture lens to get a nice bokeh effect. There is the cheap 50mm f/1.8 lens under $100 from both Canon and Nikon.
But if you do landscape, or just like to take tourist type or spontaneous photos or group shots of family or friends, then the PS camera is almost as good as the D-SLR for many casual users. There are wide angle optional lens for PS cameras that can be used for those rare occasions of large group of people.
So it is still true that your creativity and photography depend on you rather than the camera. However, the tool sometimes can limit your capability for certain tasks, but that does not mean you cannot produce some great works with a PS cameras. You just have to realize the limitation and how you can adapt and change your method to achieve the result. So if you want to freeze action in low light with PS camera, you just have to know how long your camera may take to take that shot, and trial and error to find out how to time your photo to capture the moment. It will be more difficult and takes more skill than using a $4000-5000 camera and lenses that can keep snapping away. If you spent the money, then life gets a little easier, just like many other things in life. On the other hand, having the expensive camera will not automatically improve your skill and composition (the art side of digital photography). The D-SLR may sometimes have too many distracting features that can distract you from learning the composition of the shots.
The bottom line is that if you are not going to be very serious about photography and like to take pictures mainly while traveling or for spontaneous moments, then a PS or advanced compact camera will be adequate. But if you want to get good low light action photos and curious to learn more about the different controls and settings, on the perfectionist side (and don't mind carrying the load), then D-SLR will be a better fit.