Microcontrollers/microprocessors and different bit versions, what's the difference?
It's not the width of number it can store, it's the width it can work with in a single operation.  Customarily (but not necessarily) this also has a degree of correlation to the width of native memory addressing, and thus the amount of storage which can be easily mapped without ugly workarounds such as segmentation or bank switching.  

Today's 32-bit cores are superior to 8-designs in most respects (flexibility, flat memory model, and of course performance), with the major exceptions being legacy systems, applications with extreme volume and price pressure (otherwise pricing tends to correlate better with on chip memory size than with core width), and side effects of process/density.  The later can provide things like 5v operation, or possibly in some cases greater radiation hardness or a simplicity advantage if trying to prove the CPU design itself to be free of logic errors. One final process/age side effect of value to many hobbyists is that 8-bit cores in DIP packages are common, while 32-bit devices in such packages are rarer (though they do exist).