Okay, so here's the definite answer about this thread:
Yes, PS4 is clearly graphically more powerful than PS4. On paper, the difference is about 50% - in practice, the difference is currently rather even higher: If we look at the native resolution/framerate XB1 and PS4 games are being rendered and calculate the numbers of pixels being rendered per second by each console, we're currently rather looking at a 100-125% percent graphical power difference: Because a game being rendered at 1080p at a certain framerate renders 225% the number of pixels per second than rendering at 720p and same framerate. Other games try to address the graphical power difference by picking the same resolution on both consoles, but rendering at 30fps instead of 60fps - so XB1 screenshots will look just as great as PS4 screenshots, hiding the fact that the PS4 version actually renders 200% the number of pixels per second in comparison to the XB1 version.
Now, once we can accept that PS4 is clearly more powerful, we can address the arguments brought forward by XB1 fans. There are many ridiculous arguments, but there is at least argument that indeed has a certain substance:
A theoretical 100% difference in rendering throughput CAN lead to graphical differences that are hardly perceivable for humans: Running a game at 60fps instead of 30fps requires twice the rendering throughput, but because of the characteristics of the human eye, 60fps do not feel like a 100% rendering power difference. The same goes for 1080p vs. 720p graphics: The difference between both resolutions will often look hardly perceivable, and the smaller the TV screen, the smaller the graphical difference will look.
BUT: The perceived graphical difference may actually become even more important in the near future, for example considering Oculus Rift-like technology. These VR helmets will take the industy by storm, starting in 2015. But these VR helmets need massive graphics processing power to deliver a perfect immersive experience. 60fps are said to be very important, and one must always render twice the number of images in comparison to a single, 2D screen. And when it comes to VR helmets, using inappropriate, underpowered graphics hardware will not only lead to a perceived difference in graphics quality - it will ruin the immersive experience that VR helmets are great for, meaning underpowered graphics hardware is much more likely to make people sick and vomit, just like what many people experience when watching 3D movies.
Well, even PS4 is actually underpowered for Oculus Rift-like hardware. But that only stresses my point: With new exciting technologies like Oculus Rift coming up, even the perceived graphical power difference will grow.
