I couldn't tell from the official site whether Windows Kinect is just an SDK, or another hardware piece that works only with Windows.
Yes and No
There are minor differences.
Its more like a "refresh" of the hardware. The Kinect that only supports the XBOX 360 is limited in several ways because of the hardware limits of the USB 2.0 bus interface.
So the one major "minor" difference is you can be a great deal closer to the Windows Kinect's sensor and it can still track your movements.
The second difference is the SDK released with this hardware can be used in a commercial application. The SDK that was released in the past CANNOT.
The original hardware is limited by the limitations of the older XBOX 360 models, the newest model ( released with the Kinect ) has a Kinect dedecated port, but because backware compatability is a concern the Kinect hardware itself is unlikely to be updated on the xbox 360.
Technically there is no reason the Windows Kinect hardware could not work on other operating systems.
Update: There are drivers for this device on Linux that have been released since I first answered this question.