Naziism: Political ideology
Islam: Religion.
This is why one is on to ban and the other isn't. For some reason we hold religion higher than political ideology on the scale of freedom yet they're almost identical in practice.
In Islam there is no difference between religion and politics. Your religious affiliation is literally your political ideology. That's why the Quran outlines a very specific kind of government that all Muslims are commanded by God to work towards. The fact that you'd list Islam as something separate from political ideology tells me you know very little about Islam. After the founding of the religion, it immediately began conquering the Middle East and North Africa, very violently, by the way. Gee, that almost kind of sounds like WWII, except Islam won their takeover.
The only difference between Hitler and Mohammad was that Hitler didn't claim he was a prophet of god. A mistake on his part, obviously, as he should of known all the cool warmongers claim that "God wills it".
Yes, I am aware of Sharia law. However, Islam is still considered a religion first and foremost by the world. Moreover, the Quran is more of a collection of sayings and less laws. Sharia law was developed AFTER Mohammad's death and was collected from his life experiences, his interpretations of God's word, and what he did. So if he liked cats, well... cats would be part of Sharia law. Just like how Jesus's life and actions form the basis of morality in Christianity.
And like Christianity, a ruling body was setup to write the laws and rules. In fact, both do it via proxy. The ruling bodies in Saudi Arabia and Iran are kept in power by the Islamic church but are not, themselves, the heads of the Islamic church.
The difference is not if one group has a political ideology or not (they all do once you get past the faith), it's what the ideology is based on.
If it's faith then Religion.
If it's not then "Political Ideology"
Islam is based off the belief that God backs their political ideology.
Nazisim is based off their belief that they're the best and that belief has no other backing.
Of course, I'm trying to find actual bans on being a nazi.
Several countries have bans on the symbols and salutes (with some exceptions) and the sales of nazi memorabilia is illegal but I can't find any country that outright bans the nazi party or makes it illegal to BE a nazi.