Other than the color, what exactly are the difference between the black and the white iPhone 4?
There are two changes that I've read about, both related to the color change but not just that the color is different:
The rear camera positioning has been slightly modified to reduce the amount of light that leaks into the area; it's said to be recessed slightly deeper into the device, though other reports indicate a slightly different casing. Makes sense, since the black glass and plastic would absorb light that the white glass and plastic would let through, likely causing washed-out pictures (or the software adjusting the white balance incorrectly, losing details in lighter and/or darker areas of the image).
Reports also indicate that the proximity sensor has been modified. Though it's quite possible, if the iPhone uses a capacitive proximity sensor such a change shouldn't be necessary (if it uses an infrared proximity sensor then the same light change issue would require a change). It seems more likely that the change was made for the ambient light sensor which is right next to the proximity sensor, since the ambient light sensor would have light change differences as well.
Other than the the phones are identical. The new one is no faster, for example.
Check out this gallery for some comparison photos.
Its supposed to be identical, but 0.2mm thicker
There is only a Color Difference between both devices.