Given that we're clearly dealing with a North American carnivoran, raccoon is the obvious choice: raccoons are well known for having particularly dextrous fingers that lack the sort of interdigital webbing normally present in carnivorans (Lotze & Anderson 1979). As you can see from the composite image shown here, the match for a raccoon is perfect once we compare the dentition and proportions. The Montauk animal has lost its upper canines and incisors (you can even see the empty sockets), and if you're surprised by the length of the Montauk animal's limbs, note that - like a lot of mammals we ordinarily assume to be relatively short-legged - raccoons are actually surprisingly leggy (claims that the limb proportions of the Montauk carcass are unlike those of raccoons are not correct).
More awesome rotting pics at the link.
Totally Credible Decomposing Animal Guy