What is the Difference between a Lepton and a Fermion?

A fermion is any particle, elementary or composite, that obeys Fermi-Dirac (as opposed to Bose-Einstein) statistics relating to how identical particles behave when you swap two of them. Due to an important but complicated result, this is taken to amount to having half-integer spin.

A lepton is one type of elementary particle with spin 1/2. The only leptons are the three generations of electrons, the three generations of neutrinos, and the antiparticles of these. (So that's 6, 9, or 12, depending on how you count.) There are also elementary fermions (the quarks) that are not leptons, so this is a much narrower class of particles.

A fermion is any particle characterized by Fermi–Dirac statistics and obeying the Pauli exclusion principle. So for example quarks are fermions, as are Helium-3 atoms. A fermion does not have to be an elementary particle. I'm not even sure that it has to be spin $\tfrac{1}{2}$, though I can't think of any fermions that aren't.

A lepton is a spin $\tfrac{1}{2}$ fermion that is elementary and does not feel the strong force.

So leptons are a subset of fermions. The only known leptons are the electron, electron neutrino, muon, muon neutrino, tau and tau neutrino, and their antiparticles.

The Standard Model includes 12 elementary known as fermions that respect the Pauli exclusion principle. They include six quarks (up, down, charm, strange, top, bottom), and six leptons (electron, electron neutrino, muon, muon neutrino, tau, tau neutrino) (ref)

All leptons are fermions, but not all fermions are leptons.