The advantage of a tri-fold kneeboard is the additional side pockets that sit on either side of the pilot's knee. The top surface (the middle piece in the above illustration) is a normal clipboard which holds charts and worksheets during flight. Backup documents and charts will fit in the side pockets. This model (the long version) easily holds folded aeronautical charts in the side pockets. The short version doesn't.
On the other hand, the Jeppesen tri-fold (which also holds aero charts in its side pockets) has a particularly handy way of folding up for storage, and comes with a free plotting chart.