What is the difference between kinetic momentum $p=mv$ and canonical momentum?

Regular momentum (or "kinematic momentum") is the mass times the actual velocity of the particle (for the particle case of course). Canonical momentum is that which is conjugate to a spatial degree of freedom (i.e. it is that derivative of the Lagrangian with respect to a velocity term). Canonical momentum is important for a couple of reasons but I would argue that it's greatest use is in constructing the Hamiltonian ($H = dot qp-L$, $p =$ canonical momentum). My favorite example to highlight the difference in canonical and kinematic momentum is the Lagrangian for a charged particle:

The Lagrangian for a charged particle in an arbitrary Electric and Magnetic field can be written as: $$ L = frac{1}{2}m |dot{vec{q}}|^2 - e phi + e vec{A}cdot dot{vec{q}}. $$

Of course the kinematic momentum is just $m frac{dvec{q}}{dt}$. The canonical momentum is $frac{partial L}{partialdot{vec{q}}}$ which is equal to $vec{p}=m dot{vec{q}}+evec{A}$. We use this to write the Hamiltonian: $$ H = frac{1}{2m}|{vec{p}}-evec{A}|^2 - e phi. $$ Notice that the magnetic field thus effectively contributes nothing to the energy (Hamiltonian) which is good because the magnetic field does no work on charged particles.

Now when discussing classical field theory the generalization is straightforward. Kinematic momentum becomes kinematic momentum density and canonical momentum becomes canonical momentum density. The kinematic momentum density is just the density times the velocity term and the canonical momentum density is the derivative of the Lagrangian density with respect to a velocity term. An important example to study here is E&M field Lagrangian density:

$$ L = -frac{1}{4}F^{mu nu} F_{mu nu}, $$ where F is the electromagnetic field tensor. Go ahead and try to find the momentum densities and construct the Hamiltonian as practice!