With respect to video, I suspect that the answer is that video changes with time. If a video viewer's attention drifts off of the video, he or she is likely to notice being slightly behind when he or she does glance back at the video, and therefore promptly return to task. Whereas written material does nothing active to re-attract your attention.
I hypothesize that the attention-focusing effect of full-screen reading operates via a related mechanism. Online, there are many useful and/or time-wasting attention sinks just a click, or a glance across the screen, away. Normally this kind of task switching is so common you don't notice it, so it's very easy to fall into doing it when your attention drifts. Whereas when reading full-screen, task switching requires a certain amount of effort. Also the full-screened text is a big, juicy target for your attention. Therefore attention drifts are more likely to result in a return to the task at hand.
To return to mbpershan's great-grandparent post bashing Khan for bad pedagogy and in general dismissing people for not paying attention to the learning research literature, note that I work at a nonprofit that develops well-regarded modeling and simulation based learning activities in science, and we publish in this field. So, certainly I think learning research is important. But I imagine it would be tricky to find the positive effect of video, as outlined above, in "official" learning-science research.
The effect depends very much on the fact that I sought out the video and was motivated to learn the material, and on the fact that I have the freedom to procrastinate by freely exploring the web (including interesting material I have already saved for later reading), email, twitter, other work I have lined up, etc. (If you're thinking about 14-year old kids viewing Khan videos at home, substitute Youtube videos, games, and Facebook messages as needed.) But in the official literature, often you will find something like, a classroom of kids is given some mandatory curriculum content to study--which they may have no inherent interest in--and some are selected to read paper books while others are given video with a similar presentation of the material. They're pre-tested and then post-tested and the question is asked, "were the learning gains of the video group statistically-significantly larger than those of the control (reading) group?"
Well, I wouldn't be terribly surprised to find no significant difference for the video group because the video students didn't really care that much (and were therefore happy to "space out" while the video went by, or were willing to make only nominal efforts to keep up) and because the control-group students weren't really being exposed to all the distractions of a teenager's bedroom which might tempt an otherwise-motivated student far off-task.
This is all speculation, and the whole made-up study design is obviously a straw man. So take what I say with a grain of salt. But nevertheless, learning-science results come from studies in very controlled contexts--which often have to do with mass learning of material the subjects don't choose--and often the result is remembered as a quick shorthand ("so-and-so showed that video doesn't work") that may or may not apply to any specific situation (where the question may not be about the "average" student, where self-selection and motivation may play a critical factor, and where apparently rote learning may be acceptable because the students will contextualize and criticize in later months or years the material they just learned by rote.)