The US was a bit opaque in their other reasoning, just vaguely speaking of "overly narrow scope and politicized nature" of the resolution.
In 2014 however, when a similar resolution was proposed, the EU, which only abstained had a more direct explanation:
The European Union’s member states have abstained en masse from a United Nations resolution against the ‘glorification of Nazism’.
In all, 155 countries backed the motion, which was presented by Russia on Friday (21 November) under the title “Glorification of Nazism: inadmissibility of certain practices that contribute to fuelling contemporary forms of racism, racial discrimination, xenophobia and related intolerance”.
Three countries voted on Friday against the Russian resolution: Ukraine, Canada and the United States.
Speaking to European Voice before the vote, Lithuania’s foreign minister, Linas Linkevičius, said that “no one should doubt that we are condemning fascism”, but, he continued, “under cover of this condemnation, Russia is pursuing its own agenda”.
“De facto, Russia is trying to attack the Baltic states and to determine history in its own way,” he said. [...]
Linkevičius said that it was “insulting” that a country “that is conducting aggression against neighbouring states, in a brutal way” – a reference to Russia’s invasion of Georgia in 2008, its annexation of the Ukrainian peninsula of Crimea in March and its continued support for separatists in eastern Ukraine – could put forward this resolution. [...]
The EU’s decision to abstain from the vote was also maintained by Cyprus and Greece, EU states with traditionally close ties to Russia. Most would-be members of the EU also abstained, including Albania, Iceland, Montenegro and Macedonia.
The UN vote was discussed by EU foreign ministers last Monday (17 November) at a meeting that saw a number of ministers push for efforts at the EU level to counter what is seen as a long-term campaign by Russia to use propaganda to retain influence over countries in the EU’s and Russia’s shared neighbourhoods.
“If we care about the littering of the oceans, let’s also care about the littering of minds,” Linkevičius said, noting that Russia’s “misleading messages, lies and tricks” have been “really very, very efficient”.
So it is a bit funny that countries which only abstained actually went public in condemning the resolution (or at least its motives) much more strongly....
N.B. The Jerusalem Post noted in 2016
The vote, which happens for the same resolution annually, was vetoed by the US in 2014 and 2015 as well.
In 2014 Terri Robl, who was at the time the US deputy representative to the UN Economic and Social Council, explained her opposition to the resolution, stating that the Russian government had thrown around terms such as Nazi and fascist for its own political ends.
“We believe Russia’s efforts at the General Assembly, via this resolution, are aimed at its opponents, rather than at promoting or protecting human rights,” she said.
So the US was a bit more explicit in 2014.
Also of note, the number of countries that abstained went up a bit in 2019, to 55. Also:
In 2017, [...] Immediately before the vote, the US representative proposed 20 amendments to the draft resolution, which the Russian delegation labeled "provocative" and aiming to radically change the essence of the initiative, according to UNIAN. The proposed amendments were rejected, with only Israel and Ukraine supporting the proposal.
Ukraine's representative Ihor Yeremenko said that Russia itself resorts to acts similar to radicalism, neo-Nazism and xenophobia by continuing to occupy Ukrainian territory.
Some of the "politicized" beef seems to be around this part:
The Russian draft UN resolution also expressed "deep concern" about the "increased frequency of attempts and activities intended to desecrate or demolish monuments erected in remembrance of those who fought against Nazism during the Second World War, as well as to unlawfully exhume or remove the remains of such persons."
And the US probably cannot sign up to
Member states were also requested in article 4 to "declare illegal and prohibit organizations, and also organized and all other propaganda activities, which promote and incite racial discrimination."