Why do marmosets have claws when most primates don't? What is the structure of their hands like?
Since marmosets and tamarins are considered the most primitive primates, we can't assume that evolutionary pressure led to the development of claws. Rather, it's accepted that claws disappeared as primates adapted to an arboreal lifestyle. So the question is really "Why did marmosets not completely lose their claws?" 

The clues to the answer lie in their unusually large incisors and disgestive system. The marmoset occupies an evolutionary niche where they rely heavily (20-70% of total diet) on exudates (sap, resin, etc.) from trees. Their claws, and incisors allow them to claw/chew holes in the bark to get at the exudates, while a specially adapted large intestine allows them to digest this food. The advantage of exudates is they are not seasonal, offering the marmoset a relatively stable year-round food supply. 

Other primates enjoyed the evolutionary benefits of tool use by losing their claws and gaining opposable thumbs (which marmosets don't have). I've come across no papers that describe a different phlange structure for marmosets and other primates, and there doesn't necessarily need to be one.

Most of this information came from the University of Wisconsin's Primate Research Center:

http://pin.primate.wisc.edu/factsheets/entry/common_marmoset

Edited to add (since I can't comment): One way to deduce how different features evolved is to look at phylogenetic trees. There are many out there, but I pulled this one from J.R. Roede et al. Redox Biology 1 (2013) 387-393. 



If you look at this simple tree, you'll see that marmosets diverged from humans on a time frame where it's possible they kept their claws instead of reacquiring this trait (it would be unusual, but not impossible, for this to happen). One thing you can do is dive deeper into the phylogeny, square this up with how primate hands evolved, and see if the split of marmosets from the rest of the primates happened when nails began to evolve.