Consider the Von Neumann universe $V_{\omega+\omega}$. As mentioned on the Wikipedia page on Von Neumann universes, $(V_{\omega+\omega},\in)$ is a model for $\rm Z$, but not for the Fraenkel axiom of replacement.
How can it be seen this is not a model of this axiom? I have already followed the note on the Wiki page, but was not succesful in retrieving a proof of this fact.
Look at the function $f : \omega \to \omega + \omega$ defined by $f(n) = \omega + n$. This function is not a member of $V_{\omega + \omega}$, but it's definable using parameters from $V_{\omega + \omega}$, and its domain is in $V_{\omega + \omega}$. Its range is not a member of $V_{\omega + \omega}$.
You can use this argument in more generality to show that, if $V_\alpha$ satisfies replacement, and $\alpha$ is a limit ordinal, then $\alpha$ must be a regular cardinal.
Edit: As Asaf points out in the comments, the last line is false! I'm leaving it there so that hopefully fewer people will make this mistake.