OK, simply put, the difference between a 64 bit CPU and a 32 bit CPU, in principle, is that a 64 bit CPU can calculate larger numbers right in the registers, while not having to move register contents offboard (into cache/memory) to do further calculations. (The registers are 64 bits wide instead of 32)
Fewer I/O operations = faster processing time = more complex things the computer can do in the same period of time
This may be a little oversimplified, so correct me if I am wrong.