Because it is not deflected as easily, an arrow equipped with feathers has the potential to be easier to tune. On the other hand, if you don't have contact in the first place, feather fletching is no more forgiving than plastic vanes. Some archers believe that feathers offer a much better down-range arrow stability than do vanes. Feathers are slightly thicker than vanes, and they do create a bit more drag because of the roughness of their texture, but from the data I've seen they offer only a very small advantage in stability. Plastics vanes are more durable, and easier to maintain than feathers.
Vanes are also waterproof, which can be a significant factor when hunting in wet weather.
When feathers get soggy they lay down on your shaft, greatly reducing their profile and stabilizing ability. However, there are some good waterproofing powders that will keep your feathers dry under the worst conditions. When choosing a fletching material, I suggest that you first try vanes. If you simply can't eliminate rest contact then go to feathers. Arrow building options run a wide gamut.