Data rates, data rates, data rates. A high resolution photo from a 24 megapixel sensor for example is 20 to 30MB in raw or 10MB in high quality jpeg. If you were to store that many frames for video, that would be 240MB to 720MB per second. No SD card can write that fast and the amount of information that would have to be processed by the image processing circuitry is even more ridiculous.
To give an idea of the raw data rates, a typical DSLR has either a 12 or 14 bit sensor, so at 14 bits, a 24 megapixel camera is going to produce 3642 megabytes of data per frame, that means that for 24fps video, the image processing circuitry has to be able to keep up with processing 864 megabytesover a gigabyte per second. That's almost 52over 60 gigabytes per minute or 3.26 terabytes per hour. Comparatively, for 1080p video, it only has to deal with 1/12 that information. 75MB a second is far easier to manage and with compression, that size can get down to something manageable (and keep in mind, the bigger the raw data gets, the harder it becomes to do compression in real time too).
It simply isn't possible to keep up with or store the necessary amount of video. This is also why burst mode on even high end DSLRs can only do 8 to 11 fps and why they run out of buffer space relatively quickly (typically within 20 shots or so when shooting RAW).