Many supposedly 32-bit processors are actually capable of dealing with precision greater than 32 bits. Since 1996, Intel Pentium processors with MMX (Matrix Math Extensions) technology have had 8 64-bit registers. And in 1999, Intel Pentium processors began to feature 8 128-bit registers. Additionally, coprocessors specially designed for floating point arithmetic, (such as the Intel 8087, which contains 8 80-bit registers) have existed since the 1980s. Therefore, it is a mistake to say that up until the switch to 64-bit processors, computational mathematics has been limited to numbers of 32 bits of less. Likewise, having a 64-bit processor does not imply that mathematical operations are limited to 64 bits. The AMD64 architecture, for example, has 16 128-bit registers for dealing with large/high precision numbers.
As you can see, having 64-bit registers does not mean a processor cannot be classified as 64-bit. What is important in classification is the size of the data chunks that the processor handles in a single clock cycle. A true 64-bit processor will process numbers in 64-bit chunks. Although most recent 32-bit Pentium processors have some 64-bit registers, when the processor does a mathematical operation on these numbers, it needs to break them up into two 32-bit components. Hence, it takes twice the time to process one of these numbers, compared to how long a 64-bit processor would take. It is also technically much more complicated than processing a single number.
Aside from being able to process 64-bit data chunks, a 64-bit processor will also have most (if not all) of its registers in 64-bit size. This has important implications on the amount of RAM that is supported. Pointers to RAM addresses are stored in standard registers. If a register is 32 bits, it can only address 2 32 locations, which corresponds to a little over 4GB of RAM. This means that most standard 32-bit PCs can, unless they have a hacked-together solution, only support 4GB of RAM. A true 64-bit processor is able to support far more RAM.