as megamanx00 said, under 64 bit the 32 bit programs will use more memory. Beside that, Win64 will "enjoy" more memory if it has it available, in my experience 4Gb is kind of a minimum that a 64bit installation will need to run properly. If you want to run games smoothly, I would recommend doubling that up. Memory kits are pretty cheap nowadays and you will notice the benefits immediately.
Also, keep in mind that from the very summarized specs that you published above, i can see that your video card could be the bottleneck as it is an aging model. Also, addressing the HDD could be more frequent under the 64 bit OS given that the amount of memory is kinda low. That could explain the stuttering experienced in your gaming sessions as well.
If I were you, I would consider doubling the memory> upgrading the video card> replacing the HDD. It could be that the HDD is approaching it's end-of-life, especially if it's of the same age as the rest of the system.
Hope it works well one way or the other.