Some atheists believe in right and wrong. That, however, doesn't prevent disagreement over what is right or wrong. Take lifeboat ethics. Is it morally permissible to kill a passenger to up your own chances of survival? The food and water will last twice as long with half the passengers.
Suppose two atheists both believe in right and wrong, but disagree on whether it's permissible to kill another human being in that situation. Both have a moral opinion. But they have conflicting moral opinions.
Here's the rub: given atheism, it's hard to see how a moral opinion is anything more than mere opinion.
Take a comparison: suppose there's a disagreement about the best way to treat a cancer patient. The oncologist recommends conventional cancer therapy (or perhaps an experimental treatment) while a "naturopathic physician" recommence alternative medicine. So you have conflicting opinions.
There is, however, something more than conflicting opinions. In principle, there's evidence that one treatment is more effective than another. Some treatments have a higher success rate than others. There are, of course, complications about the sample group, but it's not just a matter of opinion. There's something above and beyond conflicting opinions to underlie or undercut respective medical opinions.
In secular ethics, by contrast, there's really nothing over and above human opinion itself. You have two conflicting human opinions about what is right or wrong. But there's nothing beyond that. It's just your opinion. There's nothing additional to back it up.
There are objective circumstances and consequences, but both sides can agree on that. The point of contention is what is the right thing to do in that situation, and in secular ethics, it's just one human opinion over against another human opinion. In that case, what makes one opinion correct and the other incorrect? If all we have are human opinions about right and wrong, what makes one moral opinion true and another moral opinion false? In virtue of what is your moral opinion better than mine? Not correspondence to the "facts" of the case, for we may agree on the facts. But what makes your moral opinion a fact?