TL;DR: **One usage, one key.**

Why use separate keys in authenticated encryption?

Because the security proof of the encryption on one hand, of the authentication on the other hand, assume that their respective key is not used elsewhere.

What's the real-world cases where a system gets hacked for not using separate keys for cipher and MAC?

Assume plaintext (always one block of 16 bytes) is encrypted with AES-CBC using confidentiality key $K_C$, and then is MAC-ed (including Initialization Vector) with plain AES-CBC-MAC using integrity key $K_A$.

This is a textbook encrypt-then-MAC. If $K_C$ and $K_A$ are random unrelated secrets, and AES is a secure block cipher, and there is no side-channel, this is demonstrably secure under Chosen Plaintext Attack, and also Chosen Ciphertext Attack. But if $K_C=K_A$, then an adversary can make a forgery under mere Known Plaintext Attack (thus also under CPA).

If we detail the cryptosystem as designed:

- The sender of plaintext block $P$
- draws random Initialization Vector $\mathsf{IV}$,
- computes ciphertext $C\gets E_{K_C}(\mathsf{IV}\oplus P)$,
- computes authenticator $A\gets E_{K_A}(E_{K_A}(\mathsf{IV})\oplus C)$,
- sends $(\mathsf{IV},C,A)$.

- The receiver of alleged $(\mathsf{IV},C,A)$
- accepts the message if $E_{K_A}(E_{K_A}(\mathsf{IV})\oplus C)=A$,
- and in the affirmative deciphers per $P\gets D_{K_C}(C)\oplus\mathsf{IV}$.

Attack with $K_C=K_A$ goes:

- Intercept one message $(\mathsf{IV}_0, C_0, A_0)$
- Obtain the corresponding plaintext $P_0$ e.g. because it's no longer deemed secret after a while.
- Compute $\mathsf{IV}_1\gets\mathsf{IV}_0\oplus P_0$
- Compute $C_1\gets\mathsf{IV}_0\oplus P_0\oplus C_0$
- Set $A_1\gets C_0$
- Submit $(\mathsf{IV}_1, C_1, A_1)$ to the receiver.
- Receiver accepts the message because
- It checks $E_{K_A}(E_{K_A}(\mathsf{IV_1})\oplus C_1)=A_1$,
- that is $E_{K_A}(E_{K_A}(\mathsf{IV}_0\oplus P_0)\oplus\mathsf{IV}_0\oplus P_0\oplus C_0)=C_0$.
- Since $K_C=K_A$, that's checking $E_{K_C}(E_{K_C}(\mathsf{IV}_0\oplus P_0)\oplus\mathsf{IV}_0\oplus P_0\oplus C_0)=C_0$.
- Per the first encryption, $E_{K_C}(\mathsf{IV}_0\oplus P_0)=C_0$.
- Thus the test is $E_{K_C}(C_0\oplus \mathsf{IV}_0\oplus P_0\oplus C_0)=C_0$,
- that is $E_{K_C}(\mathsf{IV}_0\oplus P_0)=C_0$, which holds!

- Receiver deciphers per $P_1\gets D_{K_C}(C_1)\oplus\mathsf{IV_1}$ which is $D_{K_C}(\mathsf{IV}_0\oplus P_0\oplus C_0)\oplus\mathsf{IV}_0\oplus P_0$, which has no reason to be $P_0$, and thus was not authenticated (only $P_0$ was).

We can easily adapt the attack to every common block cipher mode of operation, e.g. AES-CTR. The root of the problem is reusing the same block cipher and key for encryption and authentication. We wouldn't have this issue if authentication was with an unrelated algorithm, e.g. HMAC-SHA-256.