Constructible sets of low rank can appear at later stages ($L\cap V_{\omega +1}\ne L_{\omega +1}$)

The equality $L\cap V_{\alpha }=L_{\alpha }$ can fail even at $\alpha =\omega +1$. Inside the constructible universe, $L_{\omega }=V_{\omega }$ is countable, so its definable power set $L_{\omega +1}$ is countable. The constructible power set of $\omega$ is uncountable there by the Cantor theorem. A real outside $L_{\omega +1}$ remains a constructible subset of $\omega$, hence belongs to $V_{\omega +1}$. The cardinal comparison must be made inside $L$, since its reals can be externally countable.