Kleene algebra - powerset class vs class of all regular subsets
The difference between the two is that $2^{\Sigma^*}$ contains all languages over $\Sigma$, whereas the class of all regular subsets of $\Sigma^*$ contains only the regular languages over $\Sigma^*$. So if $L$ is a non-regular language over $\Sigma$ then $L \in 2^{\Sigma^*}$ while $L$ doesn't belong to the set of all regular languages over $\Sigma$. For any $\sigma \in \Sigma$, you can take $L = \{ \sigma^{n^2} : n \geq 0 \}$, for example.