I carry my Pentax K-7 almost everywhere, so not all DSLRs live in a cupbaord. Mine is not one of the ones that will go on
eBay as 'in mint condition' - I would describe it as 'battered to chuff, but works perfectly' It's part of the reason I bought it - a magnesium chassis and weather sealing. I echo the earlier sentiments about lenses - mine almost always wears a 40-year old Pentax 50mm manual lens which is so much sharper than the 18-55 'kit' lens that it makes me wonder if the auto-focus works properly with that lens.
We also have a Canon G5 - one of their 'enthusiast' compacts, which has pretty much every setting the Pentax has, as well as having a proper optical viewfinder. Admittedly the G5 is older than the K-7, so you would expect the k-7 to be better, and it is - but the reasons why are to do with Physics.
Firstly, a dSLR has a larger sensor than almost all compacts. This means that each pixel is larger, which in turn means the sensor is less liable to generate 'noise' - the annoying little speckles you see on pictures, especially if they are taken in low light. I believe it also tends to lead to a larger dynamic range: the difference between the darkest and lightest levels at which the sensor can record details. A larger sensor also means that for a given aperture (the size of the hole that opens in the lens to let light in) you will get less depth of field (the amount of the picture front-to-back that is in focus) With very small sensors such as in phones, most things from a few metres to infinity will be pretty much in focus all the time. This effect will be less pronounced with a compact, but with a dSLR you can begin to use the depth of field by adjusting the aperture to get the bits you want to be in or out of focus. The larger the sensor (or piece of film) the more control over depth of field you have - it's for this reason that I prefer a APS-C (that's the sensor size of most dSLRs) dSLR over a 4/3rds one - but then there will be other people who will say that APS-C is too much of a compromise and one should use a 'full-frame (i.e. the sensor is the same size as a 35mm negative) dSLR - and then there are others that will say that one should use a Leaf back on a medium format body, and there are still others that say that unless you are using a 5"4" wet plate, one isn't really a photographer at all - and even then a 5"x4" is only suitable to test the composition before using a 10" x 8" In short, it's horses for courses.
The other huge advantage of the dSLR for me is that when I press the button it takes a picture almost instantly, whereas most compacts have a lag to varying degrees. I remember watching birds fly near a cliff for hours once trying to get a picture of one but the lag on my bridge camera meant I kept getting cliffs, sky and no bird.
On the other hand, the G5 is quite nice, and if I were really concerned about weight, or only wanted to record shots of where I'd been I'd be perfectly happy to use it. In fact technically it's my daughters camera and she tells me off for using it too much. But ultimately a dSLR in the hands of someone who knows how to use it will get better results more of the time - but a compact will get perfectly good results most of the time, and the beauty of digital is that you can check you're happy with the picture there and then and re-shoot if you want to.
And of course, the best camera is the one you have with you ....