Catra wants to prove to everyone she is capable, important, and even superior. She spent her developing years under constant stress of being deemed as a failure regardless of what she did. She wants a world where everything she does pays off, she wants objective validation. Shadowweaver was a scrub wounded by the fact she reached her maximum potential that was nowhere good enough compared to other mages. Nothing came easy to SW, she lived a Frank Grimes life, and saw Catra doomed to repeat that life. Hordak, what we know from our meta-knowledge, was cast aside as a failure by Horde Prime, so he is fixated on being the most powerful. We see him as leader who evaluates merits and failures, but as Catra found out, Hordak is a stone cold executioner of failures. He has no interest in the concept of learning from failure (exact opposite of Entrapta, this will pay off). To analogize, Catra assumed she was part of a Jimmy Conway gang, but realizes her boss is Tommy DeVito.
The Horde is thus composed of people struggling not just with failure, but the very real (and relateable) crisis of hard work not being good enough. Take a glance at the princesses, they all have easy power but they all just happen to get along after some deliberation. The bad guys, the Horde, are going to remain bad guys in large part to their difficulty to accept an individual relying on others. Catra has only expected Scorpia to perform her duty, but only very recently made the connection that Scorpia is doing things for Catra, personally. That was a lesson of true loyalty to Catra, and it's not a virtue Hordak ever engenders. Catra is used to operating on giving people what they want to give her what she wants, but this is fundamentally more social than Hordak's system. Sooner or later, Catra's ideas on the direction for the Horde will clash with Hordak's Horde.