Ah, I've filled out a lot of these. I've actually used both methods you describe- essay/letter type and bullet points (and gotten interviews using both styles.) I tend to favour the essay type, though (usually ends up being about 5 paragraphs or so, 600-800 words.)
The most important thing is that you address everything marked essential in the person specification and provide really specific examples from previous experience demonstrating how you meet those criteria (and if you meet any of the desirable criteria, provide examples for those points as well.) For me, it felt more natural to do that in a narrative, essay type style, but if you're not a natural writer I think bullet points can work too- as long as you have those specific examples.
edit: also, when I do my job application "essays" I'll usually break the paragraphs up with subheadings ("Personal Attributes", "Previous Experience", etc) so that it's easier to read and/or skim if needed.
edit2: regardless of which style I use though, I always have an introductory paragraph summarising what I can bring to the organisation and why I am interested in the job, as I would put in any cover letter.