 Solution

The definable power set $\mathcal{D}(A)$ performs one definability step over the single structure $(A,\in )$, whereas the constructible power set ${\mathcal{P}}^L(A)$ contains subsets of $A$ created at arbitrarily late stages of the constructible hierarchy.

For the concrete case $A=\omega$, there are only countably many first-order formulas and finite tuples of natural-number parameters, so $\mathcal{D}(\omega )$ is a countable set. In contrast, the constructible universe satisfies ZFC, and Cantor theorem makes its full power set ${\mathcal{P}}^L(\omega )$ uncountable inside $L$. Consequentlyso the two notions do not agree in general.