One likes the MC and has purer "motives" one just wants to see them burn?
This isn't the case.
Frequently, such love rivals know that the guy they set their eyes on isn't on the market, and
they don't care. If you could call such intentions "pure" (this is probably one of the least understood words these days, specifically in an anime/manga context), then they could only at most be considered purely self-serving. Even if they don't initially know, they continue pursuit despite finding out.
Women have an eye for this sort of thing: Ayase and her friend could tell that something was going on between Kurumi and Wakou; and Kurumi herself was on alert right after that encounter--she knew, on a low level of awareness, that Ayase had just put designs on her man. Like Jackist said, there's no real difference at the end of the day; it's just to create conflict, particularly for girls like Kurumi--such girls
rarely ever get a straight shot to the goal or avoid harassment in the form of such rivals once they're there.