The two approaches are not equivalent, even if they are quite similar and related. More precisely, proofs by contrapositive are a *proper* subset of proofs by contradiction.

Every proof by contraposition can be reformulated as a proof by contradiction, as you correctly noticed. Indeed, suppose you have a proof $\pi$ of $\lnot Q \implies \lnot P$; then you can prove $P \implies Q$ by contradiction, by assuming $P$ and $\lnot Q$, which yields a proof of $\lnot P$ (via *modus ponens* between the assumption $\lnot Q$ and the conclusion $\lnot Q \implies \lnot P$ of $\pi$) and then you get the contradiction $P \land \lnot P$.

But not every proof by contradiction can be reformulated as a proof by contraposition, as well explained in the accepted answer of this question. Indeed, proofs by contradiction are "more general" (i.e. they can be applied to a wider set of propositions to prove) because you can stop when you find *any* contradiction, not only a contradiction directly involving the hypotheses. More precisely, in a proof by contradiction of $P \implies Q$, we assume $P$ and $\lnot Q$ and we deduce a contradiction $R \land \lnot R$; in the particular case where $R = P$ then (usually) you can reformulate your proof as a proof by contraposition, otherwise you cannot.

*Remark.* Both proofs by contraposition and proofs by contradiction are valid in classical logic, but in general they are not valid in intuitionistic logic (roughly speaking, a constructive logic that does not admit the excluded middle law). For a more detailed analysis of the validity of these kinds of proofs, see for instance here.