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

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

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

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

does not. For example, consider $\varphi x \equiv x < \omega$.