I think the difference is that the crime is the same, but the circumstances of that particular crime must be considered as part of sentencing.
So an assault is still an assault. But it makes a difference for sentencing whether it was motivated by racism or random drunken violence. It’s about weighing up the aggravating and mitigating factors in individual crimes.
In either case, the notion of ‘hate crime’ is not all that helpful.
Hatred might not be nice, but in itself it isn’t criminal. The problem is where that hatred motivates actions and causes effects that hurt and disadvantage people.
We cannot reasonably tell people that they cannot hate (we CAN disapprove and not want to spend time with that person or listen to them). But we can prevent them from causing harm on the basis of that hate.
Frivolous example (purposefully): its fine for someone to hate broccoli. It’s not ok for them to assault someone because they put broccoli on their plate. Nor is it ok for them to try to intimidate broccoli farmers into giving up.
People can think abhorrent things and hold dreadful opinions. Where we intervene is when their actions are problematic, not just their thoughts.