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.