Although the other answers correctly explain the basic logical equivalence of the two proof methods, I believe an important point has been missed:
- With good reason, we mathematicians prefer a direct proof of an implication over a proof by contradiction, when such a proof is available. (all else being equal)
What is the reason? The reason is the fecundity of the proof, meaning our ability to use the proof to make further mathematical conclusions. When we prove an implication (p implies q), we assume p, and then make some intermediary conclusions r1, r2, before finally deducing q. Thus, our proof also provides proofs of (p implies r1) and (p implies r2) in addition to (p implies q). The proof of (p implies q) therefore provides proofs of all these other implications, and we have come to learn a great deal about the context of what is true in a matheamtical world where p holds. Basically, I am saying that a direct proof of (p implies q) tells us a great of information about what is going on in the p worlds.
Similarly, when we prove the contrapositive (¬q implies ¬p), we assume ¬q, make intermediary conclusions r1, r2, and then finally conclude ¬q. Thus, we have also established (¬ r1 implies q) and (¬r2 implies q). The proof therefore tells us about many different hypotheses that all imply q.
These kind of conclusions can increase the value of the proof, since we learn not only that (p implies q), but also we learn an entire context about what it is like in a mathematial situation where p holds (or about diverse situations leading to q).
With reductio, in contrast, a proof of (p implies q) by contradiction seems to carry little of this extra value. We assume p and ¬q, and argue r1, r2, and so on, before arriving at a contradiction. The statements r1 and r2 are all deduced under the contradictory hypothesis that p and ¬q, which ultimately does not hold in any mathematical situation. The proof has provided extra knowledge about a nonexistant, contradictory land. (Useless!) So these intermediary statements do not seem to provide us with any greater knowledge about the p worlds or the q worlds, beyond the brute statement that (p implies q) alone.
I believe that this is the reason that sometimes, when a mathematician completes a proof by contradiction, things can still seem unsettled beyond the brute implication, with less context and knowledge about what is going on than would be the case with a direct proof.