If you re-use a nonce for two different signatures with the same key, it leaks the key.

Imagine a UTXO is encumbered with a script that says “you have to use 7” as the nonce.

If you can spend that UTXO and you sign more than one transaction spending it, you leak your private key.

Lots of interesting applications (especially in multi party settings) where you can make it costly for someone to sign two conflicting transactions