The Playstation 4 GPU is 50% more powerful than the XBox One GPU.
Looking only at the hardware and setting aside the different visions for these two consoles, it's interesting how similar the components are - 8 AMD Jaguar cores (rumored to be the same clockspeed on both consoles), 8GB of memory shared between CPU and GPU, and the same AMD GCN GPU architecture. Immediately you can see how much simpler game porting will be between the two consoles, not to mention to the PC.
That said, since so much about the systems are the same, it's much easier to draw performance comparisons between the two, even before the systems are released. According to Anandtech, the Playstation 4 GPU has 1152 cores, compared to the XBox's 768. That's a significant difference, especially considering these cores are the same AMD GCN architecture, likely running at the same or very nearly the same clockspeed. Add to this that the Playstation 4 appears to have more than 250% more memory bandwidth (critical for GPU performance) than the Xbox One. This last point is likely to be offset somewhat by the fact that XBox incorporates a low latency 32MB cache for the GPU so that it relies less on main memory. Even with this addition, it's very likely the PS4 GPU will have significantly more memory bandwidth available for its 50% more powerful GPU.
If I had to guess, Microsoft tried to cut costs on the GPU/memory to give themselves room to bundle in Kinect hardware, while still maintaining a competitive price. It's a move to give the console more mainstream appeal at the expense of graphics performance.
This obviously will have some implications for visuals in games. I'm guessing for multi-platform titles, developers will target the lowest common denominator (Xbox One), and maybe spice up the PS4 version a little bit with some low hanging fruit like anti-aliasing or higher resolution textures, similar to console -> PC ports today. I think the real difference will play out in exclusive titles for the PS4, where I'm guessing after a year or two it will be very clear which platform is more capable.
In any case, this seems like a pretty big hardware difference to me. If the XBox One GPU has the power of 8 360 GPUs as Microsoft claims, then the PS4 GPU has the power of 12 360 GPUs.