There are many advantages to 64-bit and muck more than just calculation:
The additional address space. A 32-bit x86 without PAE has a maximum address space of 4GB. When you boot a 32-bit Windows system like XP it has to split the address space between hardware, kernel and application address space. Windows usually splits it so that hardware/system and application each get 2GB of address space. That means that any application cannot address and use over 2GB of memory (virtual and physical combined).
Some services which cache a large amount of data like MSSQL and Active Directory can actually run into this limitation which is why you can boot Windows with the /3G switch (which gives 3GB application and 1GB hardware/kernel address space instead). The application can then use 3GB of memory while the kernel reduces it's own caches and memory usage to fit into the 1GB along with the hardware addressing.
Under 64-bit you have a virtual address space of something like 16 terrabytes which is so large Windows actually caps it's addressible memory at a set limit like 2TB. Obviously there's plenty of addressible space for hardware, kernel and application addressing to fit without any of the hard limitations and juggling in 32-bit.
The x64 instruction set is also much cleaner and has a newer design while x86 is very old and has a lot of legacy elements that origionate from the origional 8086 and through all of the subsequent generations.
IIRC AMD-64 also adds a bunch of general purpose and floating point registers. With more of them available you can do things like store all of a functions local variables in the additional registers instead of memory which speeds up code execution.