One well known anti-Mormon has written that LDS use of the phrase "anti-Mormon" is equivalent to societal usage of the "N-word" to describe African Americans.
This is a man apparently in search of pity.
You can read his rationale here.
The problem is there is no comparison. Or is there, but not what he wants.
The "N-word" most likely derived from Americanized slang for the French and Spanish words for black, negre and negro. Today the N-word is highly offensive when used in society, (and frankly I am offended at it's use by various African-American music artists who don't seem to understand the harm they perpetuate to themselves and others by its use, but that is another discussion.) though such was not the case in earlier history. According to the 1913 Webster it was considered a vulgar application for negros; and up until the latter half of the 19th century it was not seen as insulting, per se, but rather a description of the color of African Americans as used by virtually all non-African Americans who were oblivious to the racist and arrogant assumptions incumbant upon the use of the word when applied to African Americans.
The difference is any attempt to grab the gravity of this phrase is simply a tasteless abuse of history which testifies to the suffering of slaves, brought against their will and traded as commodities, and given a name of derision.
By contrast, anti-Mormons took the title "anti-Mormon" to themselves in the 19th century. I believe even recent critics like Sandra Tanner have labeled themselves "anti-Mormons".
People bearing such a label don't suffer enslavement. Indeed, they are practically worshiped by various Christian groupies for their "love and heart" for the Mormon people. I have seen this first hand, and you can find plenty of videos on the web showing these people presenting to various church groups.
Many also get "PAID" to do this activity. And they can quit when they want. And occasionally they convert to Mormonism. Try that as one labeled by the "n-word". A slave becoming a white master, if you are going to attempt to co-op the negative connotations of the "n-word". (Yes, a strawman argument, but it just so easily fits the situation. I will try to refrain going forward.)
The term "anti-Mormon" describes something someone does, and it carries no social penalty to use the term, except among the anti-Mormons when they feel it may misrepresent their actual nature. That being said, if they are attacking the LDS faith, then their rejection of the title is more like a pedophile wanting to be called a child lover and not a monster. It does not offend the average Christian who hears the term "anti-Mormon". It does not offend the average non-Christian. It does not offend the average Mormon.
The irony is this is not a term angrily given by Mormons to reluctant recipients. It was a description of people's activities which was gladly received by virtually all such people working against the LDS faith, and worn as something of a badge of honor.
The difference between the "n-word" and "anti-Mormon" is the difference in the meaning of one who is oppressed, and one who is a racist. Those labeled by the "n-word" are involuntarily labeled and have a history of oppression caused solely because of birth circumstances. An anti-Mormon, however, is more like a racist, someone who has chosen to engage in a belief system which attacks others in the belief their system is superior.
Anti-Mormons choose to attack or engage Mormons. People labeled by the "n-word" do not seek the circumstances which bring on its application. You can be a Christian without being an "anti-Mormon", even while disagreeing with the LDS faith. I don't know of any examples where an African American has not been labeled by some ignorant bigot with the "n-word" at some point in their life.
An "anti-Mormon" is essentially a religious racist. We could call them theological skin-heads, or Christian nazi's. Each of these labels bears the implied meaning of a choice. A Christian missionary does not attack other faiths, they preach and invite into their own faith. They may or may not believe their concept of Christianity is the only way to return to God, but they don't attack other faiths when their efforts are lacking. They simply continue in their labors, and move on. Peter tells believers to be ready to give an answer for the reason for their faith, and do so with gentleness and respect. Such attitudes are usually foreign to anti-Mormons.
Anti-Mormons go out of their way to encounter Mormons in the practice of their faith. They don't preach Jesus, they preach AGAINST Mormonism, and Jesus gets thrown in like a toy in a happy meal. And that is what the word "Anti" means. To be against something. And when they say "We love the Mormon people, we just hate the doctrine", this is not true. Virtually every anti-Mormon I know attacks the integrity of LDS leaders at all levels. They attack faithful Mormons as "blind", or deceived, or even, as has happened to me, as a "deceiver". Last time I checked, these are all Mormon "people". These people basically have found a hobby which they usually support via tax free donations. I don't doubt they don't make a lot of money. They don't deserve to, so I won't argue that point. They are getting what they have rightfully earned. But they really love to try to illustrate themselves as somehow being small, diligent humble workers of Christ victimized by a huge, evil Church. It is just so untrue, it is sad.
Unlike times past when anti-Mormons chased the Mormons at gunpoint, modern society limits their attacks to attempting to ruin the religious gatherings of the faithful through verbal abuse and mocking of religious values, or simply lying or misrepresenting historical circumstances to dissuade potential converts and shatter the faith of the membership. I once heard Ed Decker defend his lies about LDS beliefs by saying something to the effect "If you're in a wrestling match with the Devil, anything goes." So for most anti-Mormons, the end justifies the means.
Boyd K. Packer recounted how he was asked "What do you think of Anti-Mormons?", to which he responded "Not very often." He explained the brethren are so busy administering and growing the Church and its affairs, they really don't have much time to spend dealing with anti-Mormon issues. I think they see anti-Mormons as just the dogs barking outside the gates of the kingdom. They make a noise, and occasionally can bite, but they really just mostly seem to do nothing constructive, leaving piles on the neighbors lawn.
Being an "anti-Mormon" means that facts are assembled in a way convenient to your conclusions, regardless of reality. They will argue things like blood atonement, God having sex with Mary, Jesus being inferior, Brigham Young ordered the Mountain Meadows Massacre or other issues without any appeal to facts. The main reason LDS leaders don't encourage members to engage anti-Mormon materials is so much of it is a waste of time. Moreover, the responses to anti-Mormons are only of value if they are found and used by those who have read the negative material. So labeling someone anti-Mormon is helpful in understanding they are not interested in growing your faith or knowledge of Mormonism, but just getting you out of the LDS Church. Examples of half-truths and denunciations about Mormons are so broadly available, providing specific examples is probably unnecessary. Answers are readily available at various locations, but I really like the FAIR Wiki for its balance of brief answers with thorough references for further research. Funny how you don't see anti-Mormon's attempting to critique the FAIR Wiki to point out its shortcomings. Why is that? I would say simply because it bites back with truth and facts to vapid attacks.
So there is no material relationship between the "n-word" and being an anti-Mormon. It is just an attempt in my opinion to wrongly attach their choices to the legacy of abuse suffered by those wrongly derided by the "n-word". Shame on those who try to make their efforts appear remotely on a par with the suffering of so many millions of oppressed people.