Strictly speaking either kashering or covering the countertop should be sufficient, however many people do both.
The reasoning for this is that kashering alone is problematic l'halacha for many reasons. First of all, the countertop must be spotlessly clean including the seam where it meets the wall. Secondly, some opinions hold that pouring hot water is only suficient to remove the flavor inparted by liquid chometz but not by a solid. Futhermore, many are of the opinion that the flow of water must strike directly on each inch of the dry surface, which is nearly impossible. It should also be pointed out that kashering can be dangerous and many people suffer burns each year while preparing for Pesach.
Covering alone presents practical issues. If one uses a thin cover such as contact paper it is very likely to tear over the course of Pesach allowing ones Pesach food to come into contact with the chomitzdig counter. Therefore, it may have become a common practice to both kasher and cover, so that the imperfect kashering will at least serve as a backup to the flimsy covering.
However, the bast solution is to use a strong thick cover such as linoleum, and then no kashering would be requred.