The first is that 64-bit applications can, theoretically, run more quickly than their 32-bit counterparts - after all, the CPU can process twice as many bits in a single operation.
64-bit processing also offers better native mathematical precision: a 64-bit binary value can represent a decimal value to around 14 significant digits, while 32 bits can represent only around seven digits.
If a 32-bit application needs 64-bit accuracy it must spread a ‘double-precision' value across two CPU registers - which, depending on the operation, can be less efficient and slower to work with.
In practice, though, neither of these technical benefits is likely to have a dramatic effect on your everyday computing. That's partly because they're only likely to give a noticeable speed boost in a few specialist scenarios that rely heavily on certain types of mathematics. One application that typically benefits is complex cryptography, so if you regularly work with encrypted data, going 64-bit could improve performance.
But applications that need to perform fast, high-precision operations typically don't rely on 32- or 64-bit processing at all. Instead they make use of extensions such as SSE4, which lets them work with registers that are actually 128-bits wide - even on a 32-bit processor.
The third difference between a 32-bit and a 64-bit operating system is the one that makes a significant difference in a real-world context. And it's the reason why, over the next few years, 64-bit processing is going to be important to more or less everyone.
Also in this series, 64-bit Windows guide:
Part 3: Beyond the 32-bit RAM limit
Part 2: Those pesky compatibility issues explained
Part 1: Should you switch?