You're not going to see any trouble anywhere - except on private property - for expressing ideas such as 'men and women differ biologically', note that no government you'd care to list here has ever suppressed speech like that.
But Nazism is on a different level, and if you're willing to go down that road protecting the free speech of Nazis you really have to be very optimistic about human nature. I keep hearing echoes of 'it can't happen here'. But I believe it can happen, and it probably can happen everywhere. The question is if we will let it and what it will take to stop it once the ball starts rolling.
They partly are. The USA, for all its faults, is the last bastion of free speech in the world. And I say this as a European.
...which is obviously tautological.
For better data, check out the democracy index: https://en.wikipedia.org/wiki/Democracy_Index#Democracy_Inde.... It's compiled by the publisher of The Economist, which is trusted rather widely, and its methodology is public.
And we are certainly not the only democratic country that have similar laws...
Dictatorship is one of the easy to find local minima of human organization, that we have to fight everyday not to slip down towards. You do that by accepting the humanity of your opponents, finding reasonable compromises, and there's no reasonable compromise to be had with the Nazi ideology, just ask the dead.
So even if it isn't a necessity it certainly will help to make it grow, and will allow it to grow faster.