The claim is that since God cannot tolerate sin in the least degree, then all sins are equal. The conclusion seems to be that we can't say anything is bad (at least, not worse than anything else) or that anything is good, or that anyone is righteous or that anyone is wicked.
I kind of stagger just reading the statement "since God cannot tolerate sin in the least degree, then all sins are equal". The scriptures speak of false Christs in the last days, and that is being fulfilled by the sheer number of people that want tell us what Christ would do without, apparently, ever having bothered to study what he has revealed about himself.
The question isn't whether this now popular argument is wrong. It's more a question of how many ways one can point out that it is wrong.
1) So first off, it directly contradicts the Savior's own words. He says that whereas most sins can be forgiven, there is one sin that cannot be forgiven. If all sins are equal, then Christ shouldn't be saying some are worse than others.
2) The Book of Mormon clearly teaches that denying the Holy Ghost is worst, next is murder, and second to that is fornication. Since the scriptures plainly state that these are the worst sins, then, well,... not all sins are equal.
3) This is just common sense. Next time someone gossips or says something rude try telling them that what they just did was as bad as Hitler trying to conquer the world and eradicate all Jews, and see how that turns out. Or maybe compare them to the people who killed the prophets in days past.
4) We experience the difference ourselves quite personally if we are living the gospel at all. We feel the difference in our own lives in the differences between our own mistakes. We feel it in the time it takes for the Holy Ghost to return. We feel it in the time we need to spend asking for forgiveness and making amends. If we know God at all, then we have experienced the difference between mistakes of different magnitudes.
5) We see this in the scriptures. The city of Sodom and Gomorrah was destroyed by fire from heaven for their sins. Imagine if fire from heaven destroyed everybody who swore on the internet. Apparently, God doesn't consider those sins to be equal. Either that, or for some reason he unjustly punished the people in Sodom and Gomorrah for doing something just as bad as everyone who swears on the internet, who he seems to be mostly letting off the hook as far as fire from heaven is concerned.
6) We see this constantly in the scriptures. Daniel, who is not perfect, was miraculously saved by God from the Lions. Whereas God did not intervene to save King Noah from being burned by fire. This seems really strange if Daniel's sins are the same as King Noah's sins.
And that kind of gets to the nub of the matter because:
7) Christ, as well as his Father, as well as all of the prophets, freely toss around words like "wicked" and "righteous", or like "faithful" and "unrighteous" like they really mean something. If Christ calls some people righteous and others wicked, perhaps we must go ahead and take his word for it that he considers some people to be righteous and other people to be wicked. Not only that, Christ seems to think we even know who he is talking about.
Yes, righteous people can fall. Yes, wicked people can repent. But both of those represent a change in the person and what they are doing with the days of their probation.
If Christ says some people, a few at least, are following the path to life, then we can believe him. We should not be correcting him and saying "No sir! All those people have sinned and come short of the glory of God, there is nothing more righteous about them than anyone else." If he says that broad is the road and wide is the gate that leads to death, then we should not be telling him "Hold on there! All sins are equal don't ya know! Those people are no worse than anyone on the so-called strait and narrow path."
The fact of the matter is that if all sins are equal than there is no point in higher covenants like the temple covenants. There is a level of commitment and obedience required to be baptized and given the gift of the Holy Ghost. But greater commitments are made in the temple. If all failings are equal, then there is no point in making greater commitments. We wouldn't really change anything by making them.
But in fact, there are basic commitments that are required to be baptized. And greater commitments required to be endowed. And still greater commitments required to be sealed. Each requires a better, more committed life.
And of course, there will be in time greater commitments beyond that until we enter our exaltation, having put aside everything that is offensive to God, and no longer partaking of even the least sin ourselves - for the least sin will have to eventually become as offensive to us as it is to him before we can say we are like him.