OK...I guess it's up to me to play a little devil's advocate here! Just for the sake of laying it all out there, and making sure all angles of the argument are fairly aired.
In general terms, a DSLR with an APS-C sensor will perform better in most situations than a P&S camera with a smaller sensor (they're all smaller - the biggest sensor available in a compact P&S camera would be 2/3...which is still significantly smaller than APS-C). DSLRs will full frame sensors (absolutely HUGE!) will outperform them all. Again in general terms.
That means that most situations, and given equal skill levels behind the camera, the DSLR will have an advantage in some of the following areas: amount of noise/grain, detail retention, lack of blooming/color fringing causing outlines around fine edges, dynamic range (as far as not blowing out highlights or burning shadows), and high ISO performance. With a good lens, the DSLR should also do better in overall sharpness/detail even when viewed large, chromic abberations (lines of colors separated from the lines of contrast...such as a faint red stripe on one side of a vertical line, and a faint green stripe on the opposite side), and purple fringing around high contrast spots.
Sounds cut-and-dry, right? Well...there are other factors. First...in good light, with the correct exposure, and a good P&S camera that has a good lens paired with it, the P&S model can perform as well as an APS-C DSLR at low ISO, even viewed at 100%. That's in noise control, color, detail, sharpness, and abberations. Alot depends on the camera and the internal processing algorithms. For example, the ultra-zoom camera I have is a Sony H5 (7.2MP). It has a miniscule 1/2.5" sensor...but is paired with a surprisingly good piece of glass - a 36mm to 432mm equivalent, Carl Zeiss, coated, F2.8-3.7 lens. A lens with those specs on a DSLR would cost you many times the price of the DSLR! On top of that, it is one of the few cameras that used 8 bpp processing for jpeg fine output, rather than the usual 3 or 4 bpp processing...so it had excellent retention of detail and fine variance in color and contrast even when cropped or viewed close up. Even Sony's newer replacements for this camera, the H9 and now the H50, have dropped down to 3 bpp processing...and the results don't seem to have been as good. In good light, I have gotten results as good, if not better, than DSLR results. I can prove it, since I also have a DSLR (10mp), and shoot with two friends with DSLRs (conveniently, a Canon and a Nikon). Both have average glass as opposed to good glass...while on my DSLR I have a pretty decent long range lens (Tamron 200-500mm F5-6.3, around $900). Blown up to full size, and compared side-by-side, the jpeg output on my H5 has exceeded in detail, sharpness, contrast, color, and control over CA/PF what came out of the Nikon and Canon DSLRs with average glass, and equalled what I can get with my 10MP DSLR and $900 lens. Of course, nailing the exposure, and knowing how to control the camera to get the most from it and not exceeding its limitations are a big reason I can get decent results from it. I've managed to sell large prints from the H5, and even get published from it in a National magazine...which do have certain resolution and quality standards that must be met. So like any other generalization, stating that a DSLR will always outperform a P&S isn't quite fair. And just as there are bad lenses and good lenses which make a huge difference how a DSLR performs, there are bad P&S models and good P&S models...and the best of them in the hands of a good photographer can match a DSLR by practically any standard at base ISO and in good daylight.
OK...back to reality. In many other ways, a DSLR is the better overall camera. Not all shots are in great light. As soon as you have to start raising the ISO, DSLRs win. As soon as you have to take an underexposed shot, DSLRs win. As soon as you have to track a moving object with high speed focusing and fast shutter speeds, DSLRs win. If you need flexibility to specialize your camera for certain types of photography, such as complex flash kit, super-low-light lenses, or specialized long distance glass, DSLRs win.
A DSLR is a much larger, more cumbersome device that can perform worse than a P&S when not used correctly, or in the hands of an unskilled photographer or paired with cheap lenses. But it is also a device that has plenty of room to grow - and the more you are willing to learn how to use all of its functions, and are willing to spend more for good lenses, it will be capable of performing at that higher level.
What I think people should consider is: What do I need from a camera, what do I want from a camera, and how much am I willing to learn from and about a camera? If the answer is an unadulterated enthusiasm for as much as you can get from it...then a DSLR is a no-brainer. But if you want lighter and smaller, if you don't want to spend percentages of your annual income, if you don't have aspirations to go professional, if you prefer a more simple interface that doesn't require learning as much...or if you just enjoy snapping some great memories here and there of your trips and events...a good P&S model will probably perform far beyond your expectations and needs and save you tons of money and learning and weight.
A DSLR which is always used with a kit lens, always used on Auto or P mode, always left in the lowest ISO, always shooting jpegs, and always used in good outdoor light, is like a hobbled athlete.
An analogy is this: Is a Ferrari better at going 55mph than a Camry? We know the Ferrari looks flashier and gets more attention, and allowed to perform to its maximum abilities at the hands of a skilled driver, it can blow away most of the traffic. But if all you're going to do is go 55...the Camry can do it for alot less money, better gas mileage, more room inside, a smoother ride, and less noise. And more reliably too! (DSLRs like Ferraris tend to like a little more attention from their owners - cleaning sensors, blowing out dust, wiping down lens contacts, etc.).
BTW, Ray - I can change the parameters of most of the modes on my A300 - I can set different focus modes, focus areas, metering modes, metering areas, DRO settings, flash modes, EV settings, etc for P mode, S, A, M, and each scene mode, so they are 'customized' the way I like them. For example - I've changed by Sports scene mode settings so that it uses spot focus instead of the default multi-point, center-weighted metering instead of default matrix, DRO on advanced mode instead of standard, auto ISO, and -1/3 EV since I like to avoid overexposure. In P mode and A mode, I prefer to use center focus, center weight metering, DRO off, ISO100, and 0 ev. The camera saves the last settings I used in those modes, so I can switch between different modes and have my personal preferences and settings ready to go.