I am not certain what you mean by the "interpretability power" either, perhaps definability? But to give you some intuition on how much "stronger" $L_{\omega^{CK}_1}$ is than $L_{\omega+1}$, here are some facts:

1. $\omega_1^{CK}$ is the first admissible ordinal after $\omega$ meaning that $L_{\omega^{CK}_1}$ is an admissible structure. An ordinal $\alpha$ is admissible iff for every $\Sigma^1$-definable function $F$ over $L_\alpha$ it is true that for all $\beta < \alpha$: if $F(\beta)$ is defined, then $F(\beta)<\alpha$. This gives you a very strong closure property: $\omega^{CK}_1$ is closed under ordinal addition, multiplication, exponentiation, etc. But taking $\omega+1$, note that not even $\omega+\omega=\omega \cdot 2$ is in $L_{\omega+1}$. So I would rather measure the strength of an ordinal $\alpha$ and its universe $L_\alpha$ not merely by how large it is, but rather by which axioms (or their weakenings) of $ZFC$ are satisfied by $L_\alpha$. As you correctly observed $L_{\omega^{CK}_1}$ does not satisfy the powerset axiom, but it satisfies (by its admissibility) the axiom schema of collection.

2. From Sacks, Higher Recursion Theory: Let $\Sigma^{\omega_1^{CK}}_1$ be the class of subsets of $\omega^{CK}_1$ which are a domain of a function $\Sigma_1$-definable over $L_{\omega_1^{CK}}$. Let $Q \subseteq \mathcal{O}$ be a $\Pi^1_1$ set of unique notations of computable ordinals. Let $n:\omega_1^{CK} \to Q$ take each computable ordinal to its unique notation. Let $A \subseteq \omega_1^{CK}$. Then:

i) $A \in L_{\omega_1^{CK}} \iff n[A] \in \Delta^1_1$.

ii) $A \in \Sigma^{\omega_1^{CK}}_1 \iff n[A] \in \Pi^1_1$.

iii) $A \in \Delta_1^{\omega^{CK}_1} \iff A \in \Sigma^{\omega_1^{CK}}_1 \land \omega_1^{CK} - A \in \Sigma^{\omega_1^{CK}}_1$.

iv) Assume $A \subseteq \omega$. Then $A \in \Sigma^{\omega_1^{CK}}_1 \iff A \in \Pi^1_1$ and $A \in L_{\omega_1^{CK}} \iff A \in \Delta^1_1 \iff A \in \Delta_1^{\omega_1^{CK}}$

v) $f:\omega^{CK}_1 \to \omega^{CK}_1 \in \Sigma^{\omega_1^{CK}}_1 \iff \{\langle n(\gamma), n(f(\gamma)) \rangle : \gamma < \omega_1^{CK}\} \in \Pi^1_1$.

There is just no way that $L_{\omega+1} \cap 2^{\omega}$ would contain all the hyperarithmetic (i.e. $\Delta^1_1$) subsets of $\omega$.

I am not certain what you mean by the "interpretability power" either, perhaps definability? But to give you some intuition on how much "stronger" $L_{\omega^{CK}_1}$ is than $L_{\omega+1}$, here are some facts:

1. $\omega_1^{CK}$ is the first admissible ordinal after $\omega$ meaning that $L_{\omega^{CK}_1}$ is an admissible structure. An ordinal $\alpha$ is admissible iff for every $\Sigma^1$-definable function $F$ over $L_\alpha$ it is true that for all $\beta < \alpha$: if $F(\beta)$ is defined, then $F(\beta)<\alpha$. This gives you a very strong closure property: $\omega^{CK}_1$ is closed under ordinal addition, multiplication, exponentiation, etc. But taking $\omega+1$, note that not even $\omega+\omega=\omega \cdot 2$ is in $L_{\omega+1}$. So I would rather measure the strength of an ordinal $\alpha$ and its universe $L_\alpha$ not merely by how large it is, but rather by which axioms (or their weakenings) of $ZFC$ are satisfied by $L_\alpha$. As you correctly observed $L_{\omega^{CK}_1}$ does not satisfy the powerset axiom, but it satisfies by an alternative definition of admissibility all the axioms of Kripke-Platek set theory.

2. From Sacks, Higher Recursion Theory: Let $\Sigma^{\omega_1^{CK}}_1$ be the class of subsets of $\omega^{CK}_1$ which are a domain of a function $\Sigma_1$-definable over $L_{\omega_1^{CK}}$. Let $Q \subseteq \mathcal{O}$ be a $\Pi^1_1$ set of unique notations of computable ordinals. Let $n:\omega_1^{CK} \to Q$ take each computable ordinal to its unique notation. Let $A \subseteq \omega_1^{CK}$. Then:

i) $A \in L_{\omega_1^{CK}} \iff n[A] \in \Delta^1_1$.

ii) $A \in \Sigma^{\omega_1^{CK}}_1 \iff n[A] \in \Pi^1_1$.

iii) $A \in \Delta_1^{\omega^{CK}_1} \iff A \in \Sigma^{\omega_1^{CK}}_1 \land \omega_1^{CK} - A \in \Sigma^{\omega_1^{CK}}_1$.

iv) Assume $A \subseteq \omega$. Then $A \in \Sigma^{\omega_1^{CK}}_1 \iff A \in \Pi^1_1$ and $A \in L_{\omega_1^{CK}} \iff A \in \Delta^1_1 \iff A \in \Delta_1^{\omega_1^{CK}}$

v) $f:\omega^{CK}_1 \to \omega^{CK}_1 \in \Sigma^{\omega_1^{CK}}_1 \iff \{\langle n(\gamma), n(f(\gamma)) \rangle : \gamma < \omega_1^{CK}\} \in \Pi^1_1$.

There is just no way that $L_{\omega+1} \cap 2^{\omega}$ would contain all the hyperarithmetic (i.e. $\Delta^1_1$) subsets of $\omega$.