Comparative writing reaches for kaffara whenever it needs an Islamic entry under atonement, and the reach is usually too quick. The two do overlap, at the point where both mean expiation, and they diverge everywhere else. Setting out the divergence is more useful than smoothing it.
Islamic teaching holds that God forgives directly. A person who repents, makes restitution where a human being has been wronged, and resolves not to return to the fault has done what is required, and no mediator stands between the repentance and the forgiveness. Kaffara sits inside that arrangement as a specific liability attached to specific faults, not as the mechanism by which anyone is reconciled to God.
The Quran also denies, repeatedly and in nearly the same words in several places, that one bearer of burdens can bear the burden of another. That statement is not aimed at Christianity in particular but it rules out the transfer on which several of the Christian accounts depend, and Muslim writers in dialogue with Christian theology have consistently identified it as the point of divergence rather than any detail of the crucifixion narrative. The Quran also does not accept that the crucifixion happened as Christians relate it, which is a separate matter and a well known one, but the deeper difference would remain even if it did.
So the honest comparison runs like this. Both traditions have expiation, meaning a prescribed act that deals with a fault. Only one of them has an atonement in the sense the Christian doctrine gives that word, meaning a single event by which the standing of humanity before God is altered. The atlas records kaffara as what it is in its own tradition and does not enter it as the Islamic answer to a Christian question, because it is not an answer to that question and the question is not asked in this form.