For the purposes of the current debate, it does make them equivalent. Morals are not absolute, but we can surely say that a certain breed of Christian hates gay people just as much as the average person hates Nazis. You may think those Christians are wrong, stupid and evil but that's really irrelevant.
So if Shopkeeper Bob hates Nazis and Gay people, why can he turn one of those away but not the other? Keep in mind that it isn't the government's place to dictate morals or interfere with sincerely held religious views.
You might argue: "Because Nazis are morally repugnant and gays are not." OK, that's your viewpoint, but Shopkeeper Bob disagrees. Why should the government reinforce your sincerely held beliefs but not Shopkeeper Bob's?
And please, I know what the law is. I'm trying to get at the underlying principle.
Originally, it was because the protected classes, which started with race and religion/ethnicity, then sex, then sexual orientation, were all groups of people who experienced widespread and pervasive discrimination. It wasn't just Shopkeeper Bob, it was a whole heck of a lot of shopkeepers and employers and others, to the point that these protected classes really found it difficult to participate fully in society. They needed the protection of these laws in order to be full citizens. Furthermore, at the time those protected class laws were passed, there was still an awful lot of discrimination, but the majority, or at least a very active minority, were able to convince legislatures that these classes were worthy of protection, and that action was needed to make that happen.
No one felt that way about Nazis. Anti-Nazi discrimination is ok with most people.
Today, I am not sure that those laws are needed. If Shopkeeper Bob has anti-gay or anti-black views, and doesn't want to bake a wedding cake for a same sex couple or a mixed race couple, the couple on the opposite end of the discrimination probably wouldn't suffer significantly, because the vast majority of shopkeepers would be offended by Bob. The couple could go elsewhere, and the backlash against Bob might hurt him more than his intended victim. Of course, this depends on the locality, and the specific circumstances.
Nevertheless, they remain the law, and they still serve a purpose, even if it is not as important today as it was in the 1950s when black people couldn't eat in most restaurants in some cities. They haven't been repealed because they do no harm, at least in the mind of most people.