Why is it so important to know whether particles are "distinguishable" or not in quantum mechanics?

What is the crucial difference between fermions and bosons?

Describe the Bose-Einstein distribution function and its implications about "condensation" of many particles into the same state.

Describe the Fermi-Dirac distribution and its connection with the Pauli exclusion principle.

Describe the connection between Bose-Einstein statistics and the nature of thermal radiation.

Describe the connection between Fermi-Dirac statistics and the electron energy distribution in a metal.