Although the Al Bhed are not followers of Yevon, unlike the Yevonites themselves, in which many of whom showed explicit hatred and prejudice towards the Al Bhed, the Al Bhed never tried to eradicate followers of the religion.
One reason why the Al Bhed are constantly ousted by the Yevonites is their use of machina, which is forbidden by the teachings of Yevon because the Yevonites believe that machina was what brought about the appearance of Sin. The Al Bhed also disagree with many of the teachings of Yevon, and at one point of the game, they kidnapped the summoners to stop their pilgrimage to obtain the Final Aeon because they did not want the summoners to sacrifice themselves to put a band-aid on the Sin problem.
The Al Bhed do not feel that Yevon's method to eliminate Sin is the best solution to the problem. If the summoner follows Yevon's teachings to obtain the Final Aeon, the summoner will die in the process because summoning the Final Aeon will kill them. In addition to the summoner's sacrifice, Sin will not be permanently destroyed, resulting in an endless cycle of summoners sacrificing themselves without a permanent solution to Spira's biggest problem.
The Al Bhed also think that the teachings of Yevon blinded the Yevonites because those who follow the religion tend to heed every word of Yevon's teachings without thinking critically if something makes sense or not, that they do not think for themselves and rely on Yevon to tell them what is true or false.
The Al Bhed's views on matters relating to the dead are also vastly different than the followers of Yevon. Yevonites believes that when a person dies, if a sending (which is believed to guide the soul/spirit to return to the Farplane) is not performed, the dead will eventually become envious and hateful of the living, which will cause the trapped soul to become a fiend. The Farplane is where those who have accepted their own death, either through a sending or by themselves, go. It is also a physical location reachable by the living. When the living visits the Farplane, the living is able to see the loved ones (who are already dead) who he/she is thinking of at the time. However, the Al Bhed believes that what one sees in the Farplane is nothing more than an illusion created by the pyreflies taking on the form of the dead person in reactions to one’s memories of that person.
Despite the Al Bhed not being followers of Yevon, they share the same goals with the rest of Spira – to bring the Eternal Calm to Spira by eliminating Sin once and for all.