I am a semi-pro photographer and I can tell you with some authority that there is much more to a $4000 to $6000 camera and it's abilities than a p&s... although in certain circumstances, some of the mirrorless, 4/3 outfits and even the little Oyumpus 4G (which shoots RAW) can take a fine photo.
Typically, the very pricey jobs not only have the capability to record a great image, but they are considerably tougher than the consumer models. I own two Nikon D810s. (about $4k each) They're great cameras, but they don't compare to the D4s which retails for over $7K. A lot of the difference has to do with the build of the body and how well it's sealed. My cameras will hold up to dust and moisture far better than say a $500 Nikon, but the D4s is damn near "waterproof"... and by that I mean you can take it out in the rain and be ok for a while, or even drop it into a creek for a moment and probably be ok.
The other big difference is the glass... the lens options... on a good DSLR vs a P&S. That goes without saying.
More expensive cameras will have better sensors (not always "bigger") that have a higher dynamic range than a $500 camera. That gives a competent photographer much more creative freedom.
None the less, many of these little cameras do a nice job and if you're shooting snaps of the kids or cityscapes, they are great. There's an old saying... "The best camera is the one that you have with you!". There are many many great images being made on crap cameras and iPhones for the simple reason that some person with "a good eye" happened to be there.
Ultimately, the camera is just a tool. All things being equal, a photographer can make a better image on a better camera. Similarly, if the photographer lacks skill, you could stick a $30,000 Red camera in their hands and they'll still take a shit pictiure.
