Use a statement when reporters are already covering a story and want quotes, or when you want to comment on a breaking story.
Statements are easier to write and quicker to get out the door.
They can be used alone or as part of a story kit -- you could pair a statement with a fact sheet and photos, for example.
Reporters at daily newspapers appreciate raw material like statements; they don't want to spend their day rewriting press releases.
If you're sending something far and wide, it's smart to take the time and turn your statement into a press release for the weekly newspapers, because they're always pressed for time.
It's easy to transform a statement into a release. Write an introduction that sums up the story and parcel out the quotes from the statement.
Statements can also be used, in some cases, instead of holding a press conference. If the person who'd face the cameras is inexperienced or otherwise vulnerable -- maybe it's a child -- you can give the media what they want without putting undue stress on the person.
Since statements are first-person, the natural language of persuasion, they're also more useful when your goal is to persuade rather than inform.
It's typical to post statements on your web page, as they're used in response to a breaking story. You don't have the e-mail address of every journalist who might be interested. They'll look for it, and when they can't find it, they'll call. Save yourself time and put it on the web page.