Αιτιολογία and Αιτιολογώ are a noun and a verb derived from αίτιο and λογος, which roughly speaking mean a set of identified causes and “reason” respectively.

So this is purely causal without imposing the person’s subjective judgement onto the action or event; simply expressing that given the circumstances, it makes sense that the events occurred, whether the person was right or justified in acting as they did is a different story. So given the events I can reason why the action occurred.

For δικαιολογία / δικαιολογώ, the words are derived from δίκαιο and λόγος again; the former meaning the person is in the right, or is morally correct from the perspective of the person expressing it. Thus, justifying is closer to δικαιολογώ.

For an example; chores are boring, so I didn’t do them even though I promised I would. You can reason why I didn’t do them but you can’t claim I was justified in not doing them. I promised I would and breaking promises is generally immoral without s good justification or a reason.

No, you (and PartiallyTyped) are right, αιτιολογώ is more or less "to explain" (but in a sense of trying to find the causes for), δικαιολογώ is basically exactly "to excuse".

Are you talking about the difference between proximal and ultimate causes? As in the proximal cause of these people dying was an explosion from a bomb set by Kaczynski, while the ultimate cause was that the things that caused Kaczynski's poor mental health that led him to those actions.

Separately I like the phrase "a reason is not an excuse" quite a lot. I can empathise with people who make terrible decisions and understand their reasoning, but that doesn't make me excuse the choices that they made.