They're the personal ones. People who don't believe gays should be allowed to be married by the state. People who don't believe women or minorities should be allowed to vote. People who believe all immigrants come here to kill/steal/crime whatever.
The difference is when it's an institution (private versus public schools for example), that's pretty easy to disagree over. One side does not believe the other is inherently evil and a 'lesser' person. When it's about individual rights, and an attack on you as a person, and your inherent rights as an individual, that is almost impossible to overcome.
I think you articulated it far better than me. Certain things I could hardly care what another person says or thinks on the matter. But on certain issues or rights if there's a fundamental disagreement, I find the opposing position not just in direct opposition, but in a way actively working to undermine my position. I can't be friends with someone when there's such a basic difference in values; if we don't agree on that, there might not be much else we agree on.