When to use it
You know something needs fixing but the ask is vague. A good problem statement describes the gap between current and desired state without smuggling in a solution. Use this before you write objectives, scope, or anything else.
A problem statement is the foundation of any project charter. It names the gap between where things are now and where they need to be, in language a sponsor can approve and a team can act on. Get it right and everything downstream (objectives, scope, success measures) has something solid to build on. Get it wrong and you spend the project solving the wrong thing.