How does a Problem differ from an Incident, and what's Problem Management's focus?

A Problem is the unknown cause of one or more incidents (often the result of several related incidents, but can come from a single one).

Problem Management proactively prevents incidents and minimizes the impact of those it can't — its focus is finding and fixing the root cause, not speed of resolution. Tightly coupled with Incident Management; if a fix needs a change, it pulls in Change Management.