A right specifically entitles someone to something, like a freedom to do something or a freedom from being made to do something. Equal rights just means granting such a thing to everyone. That does not automatically result in equal opportunity. Giving everyone the right to own a gun does not mean everyone exists in circumstances where they can purchase a gun. Giving people a right to speech does not mean everyone is in an equal position to be heard. Some people have a right, but they still don't have a plausible chance to exercise it.
In the past (and to some extent today) oppression of minorities or women has legally used loopholes in how our rights are defined to apply legal valid laws which do not directly affront the rights enumerated in the law, but have the practical effect of placing a disadvantage on the group in question and thus reducing the chance of exercising their rights.
This is really what people are arguing over with the voting rights act. You have a right to vote as a citizen (not in the bill of rights, but in the constitution, with blacks and women extended the vote in the bill of rights later), but racist state governments have in the past put seemingly innocuous requirements on voting which anyone could theoretically, potentially meet, but at a cost that weighed more heavily on blacks, thus disenfranchising them from voting. That's an example of an equal right without equal opportunity.
Equivalence is, I think, a much more abstract and vague term here. I don't usually use it in these discussion, and people use it in so many different ways that I would not presume to know what they mean. They could mean two peoples are literally identical, or they could mean two peoples are equal in merit, or they could mean two peoples are the same for the purpose of some particular perimeters they are talking about, etc...