What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proofs by contradiction, and the other proofs the contrapositive, the proofs look almost exactly the same.
For example, say I want to proof: $P \implies Q$ When I want to proof by contradiction, I would say assume this is not true. Assume $Q$ is not true, and $P$ is true. Blabla, but this implies $P$ is not true, which is a contradiction.
When I want to proof the contrapositive, I say. Assume $Q$ is not true. Blabla, this implies $P$ is not true.
The only difference in the proof is that I assume $P$ is true in the beginning, when I want to proof by contradiction. But this feels almost redundant, as in the end I always get that this is not true. The only other way that I could get a contradiction is by proving that $Q$ is true. But this would be the exact same things as a direct proof.
Can somebody enlighten me a little bit here ? For example: Are there proofs that can be proven by contradiction but not proven by proving the contrapositve?