Yes, the videos made by the Galaxy Note show more details (in this particular comparison which only included sunny outside scenes). But what does that mean? That under optimal lighting situations the DSLR from 2012 which can only do 1080p video shows less detail than a smartphone from this year which can do 4K? I could have told you that before. He could have also used a GoPro Hero3+ Black, which can also do 4K and costs half as much as the Galaxy Note.
Film makers use DSLR to make movies because of the lenses and the low light performance of the sensors, which are far better than what you will ever find on a smartphone - it is simple physics, nobody would want to carry around a smartphone which weighs 2kg or more to get the same optical performance / depth of field which the DSLR lenses allow. Yes, the DSLR makers are a bit behind when it comes to shooting video (as far as I know, Nikon is so far only considering making 4K video available and from Canon, only the obscenely expensive EOS-1DC can do 4K) - but that is because these cameras are primarily PHOTOGRAPHY devices and not video cameras.
If you'd switch the test around and made a comparison of photos shot with the DSLR and the Galaxy Note (and compared stuff like noise, distortion, sharpness in the corners of the picture, picture quality when using the built-in flash of the phone and a dedicated flash on the DSLR), you'll see that the DSLR is better at what it is designed to do than the smartphone and that there is a reason why it is more expensive.
So yes, under optimal lighting conditions, the 4K video mode of the Galaxy Note has a better resolution than the EOS 5D Mark III. It's just a bit of a pointless comparison, because it only compares one single aspect, like only comparing the acceleration of two vehicles and then declaring the faster one the better car, completely ignoring that some people might be interested in a different aspect, like ride quality, space, top speed or fuel consumption.