There are two things where the bit width of a processor matters. First there is the width of the address bus determining how much memory and I/O ports you can address - 32 bit buy you a maximum of 4 GiB, 64 bit a whole lot more, 16 EiB, that is 16 million TiB.

The other thing is the width of the registers determining the maximum size of the numbers you can process with a single mathematical or logical operation. The register width is also related to the addressable memory space because you have to store your memory addresses somewhere but this is not a hard relationship - you can stuff all kind of mapping logic between the registers and the address bus or combine several smaller registers to form a wider address - but it is quite common that the register width and the address bus width match.

Beyond more addressable memory you gain not that much. If you write your code carefully with no dependency on the bit width of the processor the compiler will happily perform the necessary transformations to make your code run even if you perform 64 bit math on a 8 bit processor. It will take a couple of instructions instead of just one, but it will work and be transparent for well written code.