Well... Banking. I'm not going to write a long explanation like other commenters, but each bank having their independent ledger and syncronizing them each X time, with lots of manual intervention is not optimal. Ditributed Ledgers allow for banks to share the information with each other and for transactions to be done instantly, without dispute issues. This technology is already being used internally within the financial industry, and it's going to stay since it facilitates a lot of internal work to the beenfit of the users.

Edit: As an extra point, these distributed ledgers are not really distributed between everyone, banks store them in secured environments and share them with other partner banks, in a controlled environment.