In this video, at 15 seconds he picks the strings somewhere up on the neck. I don't understand what he's doing. Why not just pick down below?
The closer you pluck to the bridge, the brighter the sound. The further you pluck from the bridge, the more mellow (some say darker) the sound. You can also see Clapton using the fleshy part of the finger, not the tip. This also makes for a more mellow sound. So he is using these two techniques to maximise the mellowness/darkness in this part of the song.
Clapton is plucking over the 11th fret, playing a note on the 5th fret. This is a position where there are virtually no harmonics that have a node at this point (because he's plucking at the
1/sqrt(2) point of the string, which is an irrational number that's relatively far from fractions of small numbers). As such, the sound that he produces contains the entire available spectrum (very natural sound).
He could have produced a very similar sound by playing at the point that's the same distance from the middle of the string but towards the bridge. However, the closer you pluck towards the bridge, the more pronounced the attack becomes (because the harmonics start in a phase that produces the strongest excitations exactly where you pluck), and I guess that he didn't want a strong attack in the sound of that note. This is reinforced by Clapton plucking with the berry of his thumb, which also reduces attack. Additionally, his flesh serves to reduce the high frequencies in the sound that he creates, adding to the soft, quiet character of the note that he's producing.
Guitarists frequently control the exact sound of their instrument by plucking at a certain position. Most obvious is the habit of lead guitarists to pluck at the middle of the vibrating string, and also put that middle of the vibrating string right over the pickup in use. Without overdrive, this tone sounds very clean, flutelike and soft, but with lots of overdrive it gains a lot of power to carry through the sea of noise that the rest of the band is making, while sounding very emotional at the same time. But other positions have their own uses, and I guess each guitarist has their own favorite positions and pickup combinations to produce their own trademark guitar sounds.
Where the string is plucked will make a difference to the tone of that note. Closer to the bridge makes it sound more nasal, less rich.
However, once one goes past the central position of the fretted string, the tone starts to go nasal again. Imagine plucking over fret 17 while fretting fret 5. That sound will be the 'richest', as it's being plucked at the middle point of the sounding string length.
Now, if you pluck, say, 4 frets towards the head, or 4 frets towards the body, it will sound subtly different from fret 17, but over fret 13 and fret 21, (call it 2 or 3 inches either side of fret 17) it should sound the same. I certainly can't hear a difference.
That said, there is a difference quite close to each node. Plucking close to the fret (an inch or so) on a fretted note gives a different tone from plucking the same distance from the bridge, but that's not really what he's doing.
I thought it may have been a ploy to create a harmonic, but that didn't happen.
Just one of many ways to play. We Montgomery often picked like that with his thumb. Generally, as other answers point out, the closer you play to the bridge the brighter the tone. This is because you excite a large number of high frequency harmonics of the string. The closer you pluck to the mid point of the string the warmer the tone. The guitar is an extremely versatile instrument in terms of attack and the variety of tone you can generate.
Several other answers explain things very well, but no one so far said the following, which I believe would be very useful to many:
Try it out for yourself!
Try it on electric, acoustic, and classical guitars.
Try picking the strings all the way from the bridge to the fretting finger, and see how the sound changes.
Try with the pick, with the fingertips, with the nails.
Also, do that on different instruments, and discover how each instrument has its own unique sounds -- who knows, perhaps that particular guitar that Clapton was playing at that time had a particularly sweet sound for that particular note, played in that particular way, and he was taking advantage of that.
In short: familiarize with all the possible sounds that you can get out of an instrument.
The more you do that, the more you will get an all-round sense for the full range of possibilities of the instrument, and when you will see other people doing different things, much more often than not you will know why they are doing it. And most importantly, you'll have a much richer palette of techniques at your disposal when you play.