A little oversimplified, and assuming you have enough physical RAM:
If you’re running a 32 bit app on a 32 bit OS, the OS (Operating System) uses a goodly chunk of the available RAM–your app and the OS have to share the same 4GB memory space.
If you’re running a 32 bit app on a 64 bit OS, your app gets the full 4GB of memory to itself–doesn’t have to share it with the OS.
If you’re running a 64 bit app on a 64 bit OS, it’s the same situation as running a 32 bit app on a 32 bit OS, except that now you’re limited to a 64bit memory space, which is roughly 20 billion gigabytes.
Modern OS’s “extend” RAM by “swapping” the contents of RAM onto your hard drive. This is relatively slow. The more RAM you have, the less (relatively slow) hard drive activity goes on.