Why did some Western countries, including the US, Britain, France, Denmark and Sweden vote against UN resolution 61/147 that condemned Nazism and SS glorification?
The UN resolution you refer to includes the following clause (8c):
The US constitution includes this clause (in the 1st amendment):
In the United States, the government is expressly forbidden from punishing people for disseminating their ideas, even if those ideas are stupid. Neo-Nazis are fools and brutes, but they're free to share their ideas.
(A personal note: I believe the most honorable way to defeat foolish ideas is by countering them with the truth. Punishing people for what they believe is barbaric, even when their beliefs are wrong.)
I have no specific knowledge of the history of that particular resolution. Reading over it, the wording does seem somewhat inconsistent with the rights of US citizens, described in this case by the first amendment to the constitution.
In the USA we have laws against advocating the violent overthrow of the government, and against general incitement to violence. But as long as they stay away from that line, all groups (even those that most of us find morally repugnant like neo-Nazis and the KKK) have a right to exist, proselytize, and peacefully assemble just like anyone else.
We are much more afraid of giving the government the power to decide who can or can't have free speech and assembly rights than we are about a few idiots who like to wear black and goosestep around in their free time.
A lot of the rest of the world is run by governments that do not recognize these rights, deciding who can or can't speak or assemble, judging what may or may not be said, so it may not seem like a big deal to them.