We all know which Chaos God is the primary rival of which and it mostly makes sense. Tzeentch, who represents hope, involvement, luck and direction is fundamentally opposed to Nurgle, who represents despair, apathy, cycles and stagnation. However, the rivalry of Slaanesh and Khorne is one I have a bit of difficulty figuring out. Slaanesh is the god of greed, to whom the act of wanting is the only important thing but Khorne is the god of rage, to whom the act of killing is the only important thing. Since Khorne is also a militantly dominating god it can be argued that he is the god of taking.

Slaanes takes what it wants, but Khorne takes what he wants.

It sounds a little bit farfetched to me but it's a better reason for rivalry than Khorne simply despising Slaanesh for being wussy. On such a simple level Tzeentch is far more opposed to Khorne.

Of course, it may well be that the rivalries of the gods aren't as clear as one versus the other. They might very well be each other's main rival in different fields and different ways.