I've always sorta taken this "zealot preacher" vibe with Karthus. Karthus doesn't necessarily want to hurt anybody. He'll kill people, but only to try and lead them to this better afterlife. He's by no stretch a good guy, but Karthus has good intentions, and fancies himself a sort of shepherd.
Swain takes a very different attitude. He's doing what he has to do, and he doesn't exactly care who gets hurt in the process. Swain's ambitions are conquest and power, and he schemes and claws his way to the top.
Both champions are leaders, in a way, but I think the way they go about it feels distinct enough to separate them.