They are not always equivalent. Consider the ordinal numbers. Strong induction
$$\forall x (\forall y (y < x \rightarrow P y) \rightarrow P x) \rightarrow \forall z P z$$
holds, but weak induction
$$\forall x (P x \rightarrow P x^+) \rightarrow P 0 \rightarrow \forall z P z$$
does not. For example, consider $P x = x < \omega$.