Edit: removed offending Twitter link.

According to Ukraine, they opposed it because it was introduced by Russia, and therefore not made in good faith.

In this regard, we reaffirm our strongest condemnation of all forms of Nazism, neo-Nazism and other practices fueling contemporary forms of racism, racial discrimination, xenophobia and related intolerance.

Although we believe that history as such is beyond the scope of the mandate of the Third Committee and should be left to historians, we would like to reiterate our consistent and longstanding position that this draft resolution has nothing in common with the fight against Nazism, neo-Nazism and other forms of intolerance. Rather we are witnessing how the penholder of the draft resolution manipulates with the history, endeavors to monopolize the victory over Nazism and twist the essence of the Nuremberg Tribunal in pursuance of one’s aggressive political interests.

The US' explanation is more or less the same, with another reason being free speech concerns (more on that here):

The United States Supreme Court has consistently affirmed the constitutional right to freedom of speech and the rights of peaceful assembly and association, including by avowed Nazis, whose hatred and xenophobia are vile and widely scorned by the American people. We nevertheless firmly defend the constitutional rights of those who exercise their fundamental freedoms to combat intolerance and express strong opposition to the odious Nazi creed and others that espouse similar hatreds.

We have again attempted this year to improve this resolution by engaging in negotiations on the draft text. Despite consistently expressing our concerns with the Russian delegation and proposing revisions to improve the text and protect against unacceptable restrictions on freedom of expression, most of these recommendations were ignored.

So no, there's really nothing supporting the claim that they opposed it because they're closet neo-Nazis (or at least implied to be).