What exactly is the difference between them? If I were to draw a Venn diagram, I could probably think of some contrasts. I don’t know if most conservatives really care about segregation at this point. I don’t know if conservatives are down for a genocide (though I have my suspicions). The main thing that they have in common is anything BLM or black crime related. Talk to any regular conservative about those things and they’ll turn into Derek from American History X. This is because at their core, they think the law should apply to everyone but them. They don’t care if White people get away with crimes against POC, but think POC should be executed for shoplifting.
This is a hot take, but I've come to the conclusion that in a global and historical context, there is no real distinction between fascism and capitalist liberalism. A conservative is just a moderate fascist. Liberals are slightly more progressive fascists, who disavow a desire to actively oppress those with minority identities, and usually have some class sympathies, but don't really mind watching the state grind the majority of poor people into little bits. An alt right, neo Nazi, skinhead type is correspondingly a rightist fascist. One that considers the current system to be too inefficient at enacting capitalist consolidation of all power and historical contingency. As the KPD was fond of pointing out, there are also social fascists.
I'm basing this not really on these groups self-conception of their own ideological tendencies, but simply on the basis of what material effect their ideas would have on society and what actions they are interested in taking to enact them. Revolutionary right fascists don't have to think of themselves as such to be essentially brown shirts in training.