The leader of the Social Democrats, who is hoping to replace Angela Merkel as German Chancellor, made the comparisons while speaking from his Berlin headquarters.
Both the AfD and Front National, which is led by Marine Le Pen, have experienced a rise in support as both key EU member states prepare for general elections later this year.
Hitting out at the right-wing parties, Mr Schulz said: "We will never give up our values, our freedom and democracy, our rule of law, and our plurality, no matter what challenge we are facing.
“I am saying this deliberately. Because that a US President wants to build walls, thinks out loud about torture, and attacks women, religious communities, minorities, people with special needs, artists and intellectuals with outrageous and dangerous statements, that is wholly unacceptable.
"The Front National means 'National Front' translated in German. We already had a party like this with an aggressive nationalism once in our country.
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Mr Schulz compared the Alternative for Germany (AfD) and anti-EU party National Front to the Nazis
“What a blind nationalism leads to we experienced in the first half of the 20th century.
“This is why a party of Hoeckes, of Gaulands and of Petrys is no alternative for Germany but a disgrace for the Federal Republic."
Calling for more European solidarity concerning refugee policies, the Brussels bigwig said countries, such as Hungary, would suffer the consequences if it continued to divide the Continent.
Hungary has consistently taken a hard-line against mass migration with Prime Minister describing migrants as "poison".
Mr Schulz said: "Germany is solitary when it comes to helping other countries, also financially. Germany is solidarity when other countries feel threatened and carries out sanctions.
Concerns about integration and security have pushed Mrs Merkel’s popularity down
What a blind nationalism leads to we experienced in the first half of the 20th century
“But when some members of the European Union interpret European solidarity like this: financing of agricultural funds or funds for the structure support – yes please, but solidarity for people who are fleeing – no thanks, then a future federal government has to take the issue of solidarity, when it comes to refugee policy, in the next EU budget plan into account.
"That the most vocal agent of this de-solidarisation of Europe the Hungarian Prime Minister Viktor Orban, who rejects any solidarity with Germany concerning refugee policies, is being wooed and applauded by the CSU is an open affront against the interests of the Federal Republic Germany."
Last week it emerged the former European Parliament President had thrown his hat in the ring to topple Mrs Merkel as support for her party, the Christian Democratic Union (CDU) and its sister party the Christian Social Union, has continued to drop.
The Chancellor, who has been in power for more than a decade, faces a tough re-election campaign following a major backlash over her open-door migration policy, which has seen Germany welcome more than one million migrants in the last two years.
'MERKEL MUST GO': Brussels protest against German Chancellor
Thu, January 12, 2017
Angry campaigners held placards with slogans reading 'Merkel not welcomed' and 'Merkel must go' following a spate of terror attacks against Germany.
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Protest on the sidelines of Angela Merkel's official visit in Brussels
Concerns about integration and security have pushed Mrs Merkel’s popularity down and fuelled the rise of the AfD party, which is expected to win enough votes to clear the five per cent threshold to enter the federal parliament.
Over the weekend, Mrs Merkel’s finance minister, Wolfgang Schaeuble, tried to defend the ‘open-door policy’ as he argued: “They were only human”.
Mr Schaeuble said: ”We have tried to improve what got away from us in 2015.
"We politicians are human; we also make mistakes. But one can at least learn from them."
Mrs Merkel hit out against Donald Trump as she said the global fight against terrorism was not an excuse for banning Muslims from the US.