What exactly is a counter resolution
This simply refers to the maximum count value. It depends what you are counting, but frequently it is possible to prescale the input to the counter (for example, increment it every 256 clock cycles rather than every clock). If you use a prescaler, you can still count large time intervals using a counter with fewer bits - but the precision will be reduced (the smallest interval will be the prescaled clock, rather than the original clock).

If you need a precise, non-power-of-two fraction, more count bits makes this easier. If less accuracy is needed, a 'cheaper' counter will be OK (and the manufacturer trades off the provided peripheral complexity for cost and power consumption).