Bounded Zermelo is Zermelo set theory with only bounded separation. It has the same strength as simple type theory or MacLane set theory or ETCS. It is a finitely axiomatized fragment of Zermelo, so Zermelo proves it is consistent. And Mathias proved a paradigmatic example of the difference: Bounded Zermelo proves $\aleph_0$ exists, and every $\aleph_{\alpha}$ has a successor cardinal $\aleph_{\alpha+1}$, while BZ does not prove the quantified statement "for every $n\in \mathbb{N}$, there exists $\aleph_n$."

But is there some more quantitative measure of its strength? For example, do Zermelo and bounded Zermelo have meaningful proof theoretic ordinals? I have heard that proof theoretic ordinals do not work well for theories strong enough to prove existence of power sets.