They are not always equivalent. Consider the [ordinal numbers](https://en.wikipedia.org/wiki/Ordinal_number). Strong induction

$$\forall x (\forall y (y < x \rightarrow P y) \rightarrow P x) \rightarrow \forall z P z$$

[holds](https://en.wikipedia.org/wiki/Epsilon-induction), but weak induction

$$\forall x (P x \rightarrow P x^+) \rightarrow P 0 \rightarrow \forall z P z$$

does not. For example, consider $P x = x < \omega$.