They won't. From a purely technical point of view, the PS4 is a much more powerful console. However, I think in the real world, the differences won't be quite as big as the specs suggest, certainly not on multi-platform titles. It'll be the exclusives that we'll really see the differences, and even then probably not for 12 months or so.
Its got the same CPU and if well balanced for all we know the PS4 gpu can't actually have its full power extracted because all games will be cpu limited, though in saying that the Xbox supposedly has 3 cores reserved vs 1 for the PS4(I might be making that assumption based on running one OS and the xbox supposedly reserving one core for each of 3 OS's).
Memory subsystems, again its very VERY hard to predict, having a huge chunk of insanely low latency esram on die opens up a lot of possibilities, they might use it brilliantly or in a retarded way, who knows yet.
As for theoretical power, as we've seen between consoles and PC's, huge hardware differences often don't make as dramatic a difference as we believe. Largely because design of a game is by far the biggest factor in how a game looks and the most powerful effects often have the smallest impact. This kind of thing where every higher level detail takes exponentially more power. There is no question that 1920x1080 60fps 4xaa PC versions of multiplatform games look much sharper and just much nicer than consoles, but side by side the feel of the game is pretty similar.
In general they've gone for quite different hardware that costs a different amount to make. A relatively small memory controller which uses a bit of power and then higher power gddr5 chips off die moves a lot of the power used for the memory subsystem off the main chip, making cooling and power delivery easier. You also drastically reduce the biggest die size which increases yields significantly and increases the number of die per wafer, both things make smaller dies MUCH cheaper.
Without kinect the MS chip will simply cost significantly more to manufacture. When it comes to it, it will be very interesting to see if there is a big difference, as yet we still don't know how the power will be used, people have rumoured that parts of the PS4 gpu are for compute offloading only, which could leave the normal GPU duties happening on a core much closer in power to the X1 gpu.
One of the biggest things for the consoles this gen is not going stupidly low with memory. With 8gb a piece, having memory for running multiple things in the background and being able to develop the platform, the OS and the feature set over time is going to be MUCH easier this generation.
All the talk about MS's future thinking Media features and sharing/digital distribution was mostly nonsense because they are now fully fledged PC's with enough memory that releasing essentially massively updated hugely different features will be possible on both consoles.
What the 360/ps3 could do on day one versus today, with such limited memory meant new more complex features to run alongside the old ones was difficult. With 8gb memory, adding an entirely new media centre part of an OS to replace or run alongside the current OS will be more than possible.
