I realized that this is not a less popular concern (and more popular one being the reverse) so I want to post it here. I have a Kinect for Windows but I don't have an Xbox 360 to test this, so can it work on an Xbox 360?
I own an Xbox 360 Slim bundled with Kinect for Xbox, and a Kinect for Windows. I tried plugging in the Kinect for Windows into the Xbox, and the console did NOT recognise it. So I can confirm that the Kinect for Windows will not work on the Xbox. On the other hand, the Kinext for Xbox -using the power adapter- does work on the PC.
According to Microsoft...
The Kinect for Windows sensor is a fully-tested and supported Kinect experience on Windows with features such as “near mode,” skeletal tracking control, API improvements, improved USB support across a range of Windows computers and Windows-specific 10’ acoustic models.
The sensor was specifically designed to be used with PCs, with a shortened USB cable to ensure reliability across a broad range of computers, and the inclusion of a small dongle to improve coexistence with other USB peripherals.
Kinect for Xbox 360, on the other hand, was built for and tested with the Xbox 360 only, not with any other platform, which is why it is not licensed for general commercial use, supported, or under warranty when used on any other platform.
Finally, Microsoft has dedicated a large team of engineers to continual improvements of hardware and software associated with Kinect for Windows, and is committed to providing ongoing access to Microsoft's deep investment in human tracking and speech recognition.
There doesn't seem to be a lot of information on this, but a news article on Softpedia suggests that it won't:
"Potential customers shouldn’t expect to use this version of the sensor on an Xbox 360 console, however, as the Kinect for Windows works only on computers that are running the software development kit."