Basically, the Cathars had Gnostic beliefs (any wiki article on them is a good introduction) and a priesthood that was a lot less corrupt than the Catholic priests, at least among the Perfect. They revered John the Baptist, primarily used the Gospel of John, believed that the Old Testament God was actually Satan in disguise and the New Testament God is the true High God, thereby explaining why the world and all of Creation is so bad and sinful. This belief was once held by many of the Gnostic sects of Egypt in the Late Roman Empire, the Bogomils (persecuted by the Rhomaion Empire), etc. It's understandable to see this view crop up from time to time due to the drastic differences between these versions of God. The Catholic Church tried to convert them back into their fold through an intense missionary program. In spite of their best efforts, including that of Bernard of Clairvaux, very few Cathars were willing to recant. That made the Catholic Church fear for their position of power.I've been thinking about the Cathars lately, indeed, since I've tackled on the matter of Aquitaine. I'm still unsure about what to do with them. I don't know a lot about their developments or the Albigensian Crusade. I'll need more research and some good books on them to explore a bit more their possible roles in-TL. We'll need to discuss this topic later on in better detail.
One of the reasons so few Cathars were willing to recant and why the Cathar Perfects appeared to be a lot less corrupt than the Catholic priesthood: If a Perfect lapsed (sex, meat, etc. --Perfects are required to be vegetarian and celibate), all those he “consoled” - i.e. welcomed into the ranks of the Perfects - also lapsed, and had to start over. In effect, the Cathars created an astonishingly effective method of social control for their clergy - peer pressure was very useful in keeping Cathar Perfects on the straight and narrow. Reading between the lines of Bernard of Clairvaux's reports against them, he seemed to admire them and wished that the priesthood of his own Church was as good. Women were also allowed to become Perfects, by the way, which increased their appeal to the masses while also earning more of the Catholic Church's ire.
In addition, France had very little control over the increasingly autonomous Languedoc lords and their lands which were becoming prosperous via trade. Incidentally, the Medieval cultural phenomenon of the troubadours and courtly love came from there, possibly influenced by trade contacts with Al-Andalus and their love poets. The Church and the French king promised the northern French barons the riches of the Languedoc as long as they eliminated the Cathar religious threat. Hence, the Albigensian Crusade. It was as much to destroy the religion as to bring the rich lands of the Languedoc back under northern French control. Furthermore, the Languedoc (more specifically the House of Trencavel) was nominally under the protection of Aragon but the Count of Barcelona abandoned them under Church pressure and out of fear of being invaded by the French barons while still in the midst of the Reconquista.
The microhistory book of Montaillou is a good on-the-ground look at a village in the Pyrenees after the Albigensian Crusade based on the Fournier Register. The investigation was made by Jacques Fournier in the name of the Inquisition. Fournier would later go on to become Pope Benedict XII.