Although there are certain hard and fast rules that apply to all coffee brewing methods, the answer to the question "which brewing method is best" is relative and depends on individual tastes and expectations. Coffee makers that use mesh filters, such as the plunger pot or French Press, produce a complex, full-bodied cup with a rather heavy, mildly silty mouthfeel that pleases some coffee drinkers but puts others off. Coffee produced with paper filters is higher toned, brighter, often sweeter, and somewhat lighter in body than coffee brewed with mesh filters.
Personally, I prefer coffee produced using either a good vacuum brewer or a manual pour-over drip brewer with paper filter, open at the top, which permits you to stir the grounds as you pour in the water and the brewing coffee begins to drip through the filter. In both cases the result is a full-bodied, complex cup with substantial mouthfeel, yet relatively bright, lively and silt-free.