For the same functionality, a 64 bit application will be faster than a 32 bit one - Not quite twice as fast, though, since there is additional overhead involved in processing the larger messages and commands.
Running a 32 bit application on a 64 bit OS - There is some additional overhead involved in translating between the two. Much of which can be offset by combining two 32 bit messages into a single 64 bit string - The rule being that there has to be enough room in a single 64 bit string for both 32 bit messages plus the needed tags to differentiate between the two.
That's the theory in plain english. But if I'm interpreting your poorly worded questions properly, you want actual benchmarks and definitive proof that a 64 bit game on a 64 bit operating system will be better/faster than a 32 bit game on a 32 bit OS. Unfortunately, to my knowledge there are no 64 bit games available and therefore a definitive comparison is not possible. If that answer is not acceptable to you, then I may I suggest Google as a superior alternative to Yahoo.