here are several reasons why most high resolution still cameras that also shoot video don't produce video at the same resolution, and why they can't maintain the same frame rates when shooting high resolution stills that they use when shooting video.

By far the biggest issue is the amount of data produced for a given amount of time. Since a 20MP JPEG saved with a low compression rate is typically 10 times the size of a single 1080p HD video frame, and a 20MP RAW file will be 2-3 times the size of the JPEG, that means that 20-30 HD video frames are roughly the same size as a single 20MP RAW file. To put it another way, you can store an entire second's worth of HD video in the same space as ONE 20MP RAW file.
The data coming off the sensor must be interpreted by the camera's processor. Like all other digital processors, each camera is limited by how fast the CPU can crunch the numbers. Increasing the data rate of the processor 20 fold would be prohibitively expensive, and create the need for cooling systems too large to fit in a DSLR sized package.
Current write speeds of even the fastest UHS-1 SD cards and UDMA-8 CF cards could not keep up with a data rate high enough to record 30 fps of 20MP RAW files.
When shooting still images, most cameras pause to recalculate things like focus and exposure between each frame. With a DSLR, this also requires the reflex-mirror to cycle down and back up between each frame, and the lens to open to maximum aperture between each shot. The highest end FF DSLRs can do all of this at frame rates of 10-12 fps! Lower cost DSLRs typically shoot at 3-5 fps. But they can only maintain that pace for a few seconds before the rate of transferring all of those bits to a memory card forces them to wait for space in the buffer memory to clear.
Also how do digital cameras separate the frames in video if the lens shutter never closes?

There is no mechanical shutter on the lenses of your Sony NEX-6. There is a mechanical two curtain focal plane shutter in front of the sensor assembly inside the camera's body. Depending on the settings you have selected, it normally operates as the same type of shutter does on a DSLR or even old film camera when you are shooting still images. The pixel sites on your sensor are energized just before the first curtain opens until shortly after the second curtain closes. The amount of time the sensor is energized and collecting light can be anywhere from a tad longer than the flash sync speed of 1/160 second to the maximum 30 second exposure time, or even longer when using Bulb mode.

When you are shooting video, the shutter stays open and the amount of light collected by each pixel site are read out to the processor at specific intervals, the 'counter' for each pixel is reset to zero and the pixel starts counting the number of photons that strike it again. CMOS sensors recording in 1080P (the p is for progressive) read each pixel site from the top to bottom of the frame in sequence, then start over at the top for the next frame. Older Standard Definition and 720i cameras read the odd numbered lines from top to bottom, then go back and read the even numbered lines (The i in 720i stands for interlaced). Higher end CCD sensors read the entire sensor at the same instant, dump that data to an on sensor buffer, and begin collecting more light while the data from the previous frame is sent to the camera's processor.