Positive lightning is indeed very interesting. According to the National Weather Service, the reason why it has a higher amperage and voltage is due to the due to the amount of air it must get through to reach the ground. Otherwise, it would never reach the ground.
Since it originates in the upper levels of a storm, the amount of air
it must burn through to reach the ground usually much greater.
Therefore, its electric field typically is much stronger than a
negative strike. Its flash duration is longer, and its peak charge and
potential can be ten times greater than a negative strike; as much as
300,000 amperes and one billion volts!
This would also account for the brightness of positive lightning and the extended duration, as a higher amperage & voltage strike would appear brighter and take longer to dissipate.
This page talks about why positive lightning occurs. It says:
"Positive lightning" is a simple process of completing an electrical
circuit from the cloud to the ground and back to the cloud. As
negative lightning strikes the ground, the ground becomes negatively
charged over a wide area. As the storm moves over this area and past
it, the H+ in the upper reaches of the cloud feels this potential
difference from the ground and sends out a positive streamer which,
for all intents, looks like normal negative lightning. The ground
sends up a negative streamer and once these two join, electrons from
the ground flow upwards to the H+ in the top of the cloud. The ground
then becomes neutral once again. The ground will hold the negative
charge of dozens of negative strikes, building up a huge potential,
before positive lightning will occur. This is why positive lightning
is so powerful.
Positive lightning is probably so rare because it takes so many negative strikes in order for enough energy to build up to cause a positive lightening strike.
There are quite a few interesting facts about positive lightning:
Some interesting properties of positive lightning:
Positive lightning can be the dominate type of cloud-to-ground during the winter months and in the dissipating stage of a
Positive lightning has been identified as a major source for the recently discovered sprites and elves. Sprites and elves are most
likely lightning discharges but occur from 18-60 miles (30-95 km) in
altitude, well above the parent thunderstorm.
Positive lightning is usually composed of one stroke (negative lightning typically consists of two or more strokes)
Another NOAA site says that positive lightning can strike "near the edge of a cloud", or even 10 miles away from the cloud! This means that a positive lighting strike could occur somewhere where the sky is blue, and it's not raining.
In addition, there is also a type of lightning called "bipolar" lightning, which changes its polarity.