Why are helium-4 nuclei considered bosons, while helium-3 nuclei are considered fermions?
From the Wikipedia page on Identical Particles:
Examples of bosons are photons, gluons, phonons, helium-4 nuclei and all mesons. Examples of fermions are electrons, neutrinos, quarks, protons, neutrons, and helium-3 nuclei.
How does the difference of one neutron change whether it is a boson or fermion?
Bosons by definition have integer spin, while Fermions have half integer spin. Neutrons and protons are Fermions, but $He^4$ has both neutron and proton spins oppositely aligned so it's total spin is 0 (hence a Boson). $He^3$ on the other hand has the proton spins oppositely aligned (total spin 0) but the remaining neutron has spin 1/2 . Thus the spin of $He^3$ is 1/2 (Fermion).
Protons and neutrons are spin 1/2 particles. 1) He3 and He4 both have 2 electrons. These electrons form spin singlet state. 2) From the nuclear shell model, the two protons form singlet in '1s' shell, also for two neutrons. Therefore, the nucleon of He4 is spin 0+0=0, which means boson. 3) On the other hand, He3 has only one neutron, so only one neutron occupies 1s shell. Therefore He3 becomes spin 1/2 particle due to the neutron spin, which means fermion.
