However, if you were to ask what is the difference between an Absolute Monarchy and a Dictatorship, I don't have much of an answer. Those seem much closer together.

For purposes of the original claim that monarchies are more stable, I think we can ignore constitutional monarchies, because those are just vestigial and are completely different from the other kinds.

Non-absolute monarchies (where nobles or a non-popularly-elected assembly share power) do count. It would help at this point to see a quantitative review of monarchies (absolute and non-absolute) and compare their stability trends with dictatorships.