[jsutton]

There's been a few examples of use cases said by others in this thread, and your financial settlement use case is a great one. Any time you need to move money around multiple parties based on a clear set of rules or criteria, blockchain could be useful. Insurance claims, international remittances, lending, to name a few. Anywhere a valuable process is still dominated by paper (and there are surprisingly many).

Generally, being able to program the management of money (i.e. if personA meets a certain criteria, send $X to personA) in a transparent system can be incredibly powerful.