(Answer originally posted on Quora)P4 (All-fourths) and M3 (All major thirds) tuning both have the huge advantage of being regular tunings. To this list we could add P5 tuning (All-fifths), and I also know about a tritone tuning as well.
Standard tuning = Inconsistent tuning
The standard tuning of the guitar (EADGBE) is based on fourths, however, there is an inconsistency between the G and B string ; they are a major third apart. This inconsistency makes it harder to visualize intervals on the neck as they are not consistent across string sets.
Using a regular tuning means that all fingerings for scales, chords, arpeggios, or whatever else, is movable both along AND across strings. A same fingering will always produce the same intervals.
This is immensely helpful when trying to improvise or compose music. The connection between mind and fingers happens way faster on a regular tuning than it does on standard tuning. Intuition develops faster. The brain just knows how to grab the sound it is hearing without giving it a second though since the interval map becomes so predictable.
Less rote learning, faster learning
A chord shape on standard tuning has to be relearned 3 times. A simple triad in root position has a first fingering when it starts on strings E or A, then an other when it starts on the D string and yet an other when it starts on the G string. On a regular tuning, a same chord shape would do the job anywhere on the neck.
As a consequence, the rate of learning is way faster on a regular tuning than it is on standard tuning.
As an improvisation tool
Jazz players completely benefit from the advantages offered by regular tunings as they are mainly concerned about improvising and developing that mind-fingers connection as fast as possible.
They also suffer few of the drawbacks. The loss of repertoire (songs designed with the standard tuning in mind) is the main one, but since playing jazz standards do not call for exact replicas but is instead open to interpretation, they simply do not care that much.
Various regular tunings
P4 tuning is the closest regular tuning to standard tuning, which makes the transition easy. My guess is that it is the most used regular tuning.
M3 tuning makes position shifting almost completely avoidable since there is only 4 frets (one by fingers) before we need to string shift. It is sometimes played on 7-strings guitars to compensate for the loss of range.
On the other end, P5 tuning requires big stretches on the fingers. It is the same tuning as violin and viola players. It also extends the range of the guitar significantly.