I see another answer is already accepted but I feel like throwing in my two cents anyway.
For references, I know the final story in Asimov's I, Robot collection dealt with computers making important economic decisions, and gently "shaking the boat" by fudging some numbers so that people who disagreed with them would be removed from power. (It sounds ominous when I put it that way. It wasn't like that in the book.) He didn't go into the specifics of technology, but they were implied to be a half-dozen or so supercomputers scattered around the world, and ran on the same magic positronic brains as the rest of Asimov's robots.
Another Asimov story, The Last Question, had an AI running the entire universe after all the humans populating every galaxy were put into a Matrix-like cryostasis so that they would never die. (Geez, were all of his stories so dark and I never noticed?) This one was even more handwavy-magic than the positronic brains. He even says that it cannot be comprehended by mere humans.
On a lighter note, the excellent webcomic Schlock Mercenary has had many AI characters in government positions. They are basically depicted as really smart people, and perform all the functions of the government. They run on magic quantum ternary matrices, but as far as I know neither quantum nor ternary have any advantages for AI. (My personal opinion is that the randomness of quantum mechanics could mirror the chaotic nature of neurochemicals, but I have no idea what I'm talking about so ignore me.)
To answer your actual question, the benefits and drawbacks depend heavily on what kind of computer, how advanced, etc. If it's basically a mind like ours in a metal shell, but faster, it could probably run all the government's functions and handle the bureaucracy more efficiently than any group of people ever could. The biggest danger would be it getting bored and going off to do something else. Such minds are obviously way beyond our current technology, and it's anybody's guess what technology would be required.
On the other hand, if it's just making economic decisions (production, distribution, etc.) a modern computer could probably handle it. We already have supercomputers trading stocks thousands of times per second. With the right algorithms, it could run the military as well: consider that computer games have enemy AI, and these days it's sometimes even smart. Evolutionary algorithms and machine learning could overhaul our tactics, without the need for any technology beyond what we have today. But such "decision" machines are not something I would trust to handle lawmaking, policing, judging, or any other government functions that require some understanding of impacts on human lives.