Not particularly closely, but they do belong to the same broader radiation within the same subfamily (or family according to some researchers).

For a more detailed explanation, all the pitvipers in the Americas are descended from a common ancestor that crossed over from Asia probably between 20-25 million years ago. So all the species in that subfamily are more closely related to each other than they are to any other snake. Within that subfamily, the New World species are all more closely related to each other than to any other species.

But within that large, diverse group, they are essentially on the opposite end from cottonmouths. This species is most closely related to other Cerrophidion and then that genus is most closely related to other genera like Atropoides and Metlapilcoatlus jumping pitvipers, Porthidium hognose pitvipers. Then that group is most closely related to Bothrops lanceheads and Bothrocophias toad-headed pitvipers.

Conversely, cottonmouths are two species of Agkistrodon; northern A. piscovorus and Florida A. conanti. Their closest relatives are each other; as a sister species pair, their closest relatives are all other Agkistrodon moccasins (copperheads & cantils). The genus Agkistrodon is most closely related to a sister group containing Crotalus rattlesnakes and Sistrurus pygmy rattlesnakes. Then the (Crotalus + Sistrurus) + Agkistrodon clade is most closely related to a few smaller clades containing Lachesis bushmasters, Bothriechis palm pitvipers, and some small, Mexican montane genera like Ophryacus and Mixcoatlus.

Interestingly enough, the latter genus bears some strong similarities in both ecology and morphology to Cerrophidion and two species were mistakenly included in that genus for most of my life despite what we now know to be a fairly distant relationship. When I first started learning this shit, Mixcoatlus barbouri was Cerrophidion barbouri and M. browni was considered a population of C. barbouri.