The dynamic range tends to increase considerably and I believe another benefit might be the decrease of the noise floor/increase in signal sensitivity but I could be completely wrong on that second aspect. With respect to SDR hardware it's not a measurement of processing power like it might be on a computer - it's a measurement of dynamic range because the bits being handled are done by the ADC (analog to digital convertor) so it's more a measurement of audio quality in the long run than anything else.
So just to clarify: an 8 bit SDR device may sound a bit worse than a 12 bit SDR device when tuned to the same frequency and not have the dynamic range the 12 bit ADC is capable of, and it might even offer improvements with respect to tuning in digital modes as well because of a cleaner higher resolution signal.