The moles who worked on the Manhattan Project varied in their skills and thus influence. All worked to aid the Manhattan Project, but they also worked to aid the USSR. The two were not in any way mutually incompatible; the USSR and USA were allies against fascism, and a major motivation for many of the spies was more about sharing the knowledge gained in a mutual fight against evil as opposed to aiding the USSR at the expense of the USA. They were also not, for the most part, "taking orders" from Moscow so much as volunteering information on what they were already working on — they were "moles" rather than "trained spies" for the most part (the "trained spies" tended to help funnel the information from the moles back to Moscow).

To give a very basic overview of the main ones: Klaus Fuchs was immensely important to the project. He helped developed the theory of gaseous diffusion used to enrich uranium (his name is on the patent for it!), he was intimately involved in the calculations used to develop the implosion weapon, and he helped develop the theory of blast waves used for targeting purposes. Along with John von Neumann he also developed an early H-bomb design whose value was only appreciated much later. The fact that he was a spy shocked the rest of the scientists when it later (in 1950) was revealed, in part because he had been so trusted and so close to the center of things, and he was a friend and collaborator with some of the top people on the project (Hans Bethe, John von Neumann, Edward Teller, etc.).

Theodore Hall, by comparison, was a much lower-level person, though his work was still important. He was the youngest scientist at Los Alamos and helped with math and physics issues relating to implosion and a few other things. Unlike Fuchs there isn't really anything you can point to and say, "oh, that's largely Hall's work" — he was a useful worker but not a star or anything like that.

David Greenglass was an engineer in the Special Engineer Detachment. He helped set up things that the scientists wanted done. He was not a gifted scientist or anything like that. He had better technical skills than your average grunt, but he was mainly there to serve as a form of raw labor. He could have been replaced by thousands of other people; his individual contributions were minor.

George Koval was the only trained agent that we know about who was also a technical person on the project. He is interesting because his contributions to the project were supportive — he was a health physics officer. This gave him broad leeway to see and learn things that might have otherwise been classified or kept from many other people, including about some of the important processes involve with uranium enrichment, plutonium production, and polonium production. His contribution to the development of the bomb was in many ways quite minor in this respect, at least from the point of view of building it, but it gave him a lot of access.