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