This is one of the most frequently asked questions. There is basically no general rule simply because everything depends on application purpose and which CPU features it will use in work.
When we compare 32-bit and 64-bit software in their native environment (32-bit or 64-bit operating system), 32-bit applications usually use less memory than the 64-bit equivalent.
This is because 64-bit software is created to work with different architecture and it uses 64-bit values in production. At the end, this might cause a 64-bit application to use much more memory.
32-bit application runs faster
This means that the 32-bit application will run faster than the 64-bit equivalent. The extra memory usage affects the program startup and shutdown times and all other activities that involve accessing your drives.
64-bit application process data faster and improve stability
However, thanks to the 64-bit architecture, native applications can access new and faster features of 64-bit CPUs. This might result in 25% better processing speed and a better overall software stability.
A 64-bit OS can work with large amounts of memory. Because of that, a 64-bit system can be more responsive when running several programs at the same time as well switching between then more frequently.
32-bit Applications on 64-bit OS
Most 32-bit applications are supported by the 64-bit systems. Once the 32-bit application is running on 64-bit Windows, it requires WOW64 emulation to run. So any 32-bit performance is not noticeable on the 64-bit operating systems.