I read on Wikipedia that the circle of fourths (as opposed to circle of fifths) is preferred in the analysis of Jazz music. Why is this so? Are the any good examples to explain this?
It's essentially just a matter of perspective. The circle is only organized differently for different purposes.
The circle of fourths and circle of fifths are, in fact, the same thing, but written in different directions. This is because P4 and P5 are inversions of each other. For example, a G going to C could either go up a fourth or down a fifth.
Both Classical music and Jazz are also largely based around the same basic V-I chord progression. Jazz uses this in countless phrases built around ii-V-I progressions. (Note that ii to V is also an ascending fourth). Classical extends this pattern even further in many harmonic sequences, which proceed as iii-vi-ii-V-I. These progressions can all be analyzed as either the root of the chord going up a fourth (clockwise on a circle of fifths), or going down a fifth (counterclockwise on a circle of fifths).
One good reason for using the "fourth-wise" version of the circle is that going around clockwise will always give you the next chord in one of these fourth-based progressions. Thus its more practical for determining chord progressions.
The reason for initially writing it in the "fifth-wise" direction is that its formation was rooted in the recognition of related tonalities (not for organizing chord progressions). The relationship of the fifth was a better interval theoretically, because of its use in Pythagorean tuning, triads, half cadences, and the idea of related keys.
TLDR: Jazz asks "Where do I go next from here?", while Classical music asks "Given that I want to end up here, where do I have to come from?"
It makes relatively little difference, but it is preferred by some because the more standard direction of tonal chord progressions will move clockwise in a circle of fourths, and they see that as more natural.
Example: A very common chord progression in C Major is Dm—G—C, or ii—V—I. In the more common circle of fifths construction, this is moving counter-clockwise from 2-o'clock, to 1-o'clock to 12. In the circle of fourths—which is ultimately just a mirror image of the circle of fifths—this will be clockwise movement from 10-o'clock, to 11-o'clock to 12.
So it's ultimately just different ideas about which presentation of the same idea is more intuitive.
I'm not aware of this as a jazz vs. classical distinction, though that may have some truth behind it. I first encountered the circle of fourths in a textbook aimed at students of classical common-practice theory.
Using a section of the circle of fourths:
G C F Bb Eb
You can see the 7th to 3rd progression, which is very common in chord progressions and improvisation. The classic 2-5-1 in theory follows the circle of fourths. Resolving the 7th to the 3rd of adjacent chords gives a very nice movement to songs and solos.
Gmi C7 Fmajor (2-5-1)
2-5-1, 3-6-2-5, are great structures and are built on the circle of fourths.
I find that nearly every genre of music uses the fourths progression. I draw the circle out into a vertical list for ease of playing. The proggresion of chords is upwards in fourths. The normal pattern is- form the root chord (FOR EXAMPLE C), the one above is used. Then the sequence will jump down the list, a number of steps, then return to the root chord through the fourths sequence. Example with four chord jump. C, F, C,JUMP A, D, G, C. This is a basic pattern. Of course, the great joy in harmony, is the break this sequence in many ways. There many common variations, but quite often, the sequence is not used at all. It is very educational to take a buskers chord sequence, and follow it on the fourths chart. This shows the pattern very well, and is a great way to learn for 'by ear'players like me. You also change the key of a piece by simply moving it on the chart. Mick Reeves