This is a follow-up to a question I asked earlier over here. Why are stationary limits so ubiquitous when studying large cardinals?

I have noticed that there appears to be a stronger limit notion that appears sometimes - a "measurable limit". From what I understand, a cardinal $\kappa$ is a measurable limit of cardinals of type X if there is a normal measure on $\kappa$ containing all the cardinals of type X $< \kappa$

From this definition, it is unclear to me why a measurable limit is stronger than a stationary limit. For instance, is a measurable limit something like a stationary limit of stationary limits ? Or something much stronger ?

What would be examples of theorems illustrating why a measurable limit is stronger than a stationary limit ?