If a DSLR can record 1080p video at 60fps, why is it only capable of continunous shooting at, maybe 5fps? If frames per second are frames per second why can't you take 60 photographs per second? Is there something fundamentally different about a frame of video, compared to an individual photograph?
Let's say the average DSLR shoots at around 10MP (and that's probably a low estimate these days.) The number of pixels in a 10MP photo is around 3872 x 2592 (assuming a somewhat standard sensor.) Clearly much larger than your 1080 x 1920 video. "10MP video" would be in the 4k range. Which by the way is all the rage these days. :)
There's also the fact that video systems tend to use interlaced field capture where basically every other line is scanned and the rest are filled in from the data around it. You don't really notice this in the video but if you take a still from a video it's pretty easy to notice the lack of sharpness.
Hope this helps.
A still frame from a modern camera is 15+ megapixels, and potentially going to have every pixel looked at with a critical eye. A frame of HD video is 2 megapixels, and not going to subjected to the same level of analysis at 100% crop; at this point, I think it's more a question of why you'd think they would have the same frame rate.