Paper as a filter medium: Everyone always says paper wicks away oils and flavors. What you need to know is that rinsing paper filters help but many paper filters are just going to taste like paper. To really understand this, you need to stop drinking coffees with paper filters for a while and then come back to it. The contrast really makes it clear.
The other thing you need to know is that fines will build up against the walls of the filter (subject to the shape of the filter). Cumulatively, we describe the tastes from paper as thin, brighter, and crisp. Look for paper filters that are less thick and avoid brown filters.
Metal as a filter medium: Metal mediums (much like glass) depend highly on the hole sizes (and their spacing). Amounts of fine particulate are expected to pass through to the resulting cup. The effects in taste notes are simple, regardless of what form factor. This filtration that lets sediment through will have more body, more texture, less clarity, and can result in an over extracted harsh taste in the cool cup (from sediment). The fine particulate should be avoided and compensated for in methodology as much as possible.
Cloth as a filter medium: Cloth is unique and a bit unorthodox these days. It is the only filter medium in common use that has natural depth filtration. The fines will be trapped effectively and oils will pass through resulting in a heavier cup with great clarity. Cloth is hard to care for and can easily impart off flavor notes if not rinsed properly after use, cared for in between uses, and cleaned properly as needed. It also requires a seasoning brew the first time it will be used! That's a lot of work so we often recommend cloth for the diligent or advanced user.
Sidebar: Surface Filtration vs Depth Filtration + Cake Filtration
Surface filtration is where the filter medium keeps all particles, that do not pass through at the surface of the filter, on the surface. Depth filtration allows for particulate to pass through the surface layer and into the depth or layers of the medium. This means that the finest particulate is less likely to build up in a layer on the surface and is captured in that depth.
Cake filtration happens in the brew cake of a pour over (espresso pucks as well). As the grinds settle in a brew cake, the layers of grounds act themselves as a filter. This extra bit of filtration is dependent on two things: depth and agitation. Two good examples of cake filtration are the narrow top chamber of a Syphon and the shallow wide brew puck in a Clover brewer. Unlike a Syphon, the brew chamber design in a Clover has a shallow puck with very little cake filtration. (However, by whisking or strong late agitation, a brew cake will not form and have little to no effect on filtering the resulting brew of a Syphon).
This is why we push for less circular stirring agitation in Abids/Syphons/Clover, less pours around the edge of the brew cake in hand pour, and less aggressive pouring (lower steady flows are better). To achieve more clarity and layers of flavor in the resulting cup, a focus on achieving cake filtration seems essential.
We recommend paper if rinsed well for the every day user, cloth for the advanced or technical focused, metal filters for those who are looking for most ease of use. Cloth is our favorite in shop as it works extremely well for the volume we have. Some time ago while in Japan, I played around with and got a Cafeor (a metal screen filter v60) from Hario which is similar to the K-One (has larger holes spaced differently). We found that it worked reasonably well with it's screen configuration where the screen was only about 60% of the surface area but the sediment was still a problem. We adjusted our method quite a bit to compensate for the sediment and found a coarser grind, longer pre-infusion bloom, and dumping the first drops during pre-infusion helped. The sediment though proved for a often unpleasant cool cup and we much preferred the results we get in Syphon and Cloth hand pours.
Aside from the Cafeor, our trainer and an engineering friend played with the new K-One. Neither liked the results and were caught off guard by the enthusiasm others had when tasting the same cup. Regardless, we are going to play in shop with one tomorrow care of Dylan Evans who is gonna bring it in to give it a full working over. I believe our experiences with the Cafeor will hold true. We've tested everything and choose to stick with the methods we like the most regardless of how difficult or complex they get as long as we can find a way to execute consistently with them on bar.
In the end, choice of filter medium is really about a match of personal taste, patience for details, and ease of use. Try them all and work until you find what you like.