'Point' particles become fuzzy and subject to a wave equation
One must not confuse the wave equations of classical mechanics with the wave equations of quantum mechanics. Mathematically they have sinusoidal solutions, but it is the variable assignments on the mathematics that describe measurable physical quantities.
The classical solutions of Maxwell equations describe the variations of energy in space and time, the Poynting vector and the power carried by the classical electromagnetic waves.
The solutions of the quantum mechanical wave equations do not describe energy transfer and are chosen by the postulates of quantum mechanics, , basic of which is that the wavefunction
It is the probability that waves, as seen in the double slit experiments one particle at a time
The same behavior is seen for photons .
Photons wavefunctions obey a quantized maxwell's equation.
Then, if we have charged particles interacting through the electromagnetic field, is there an essential distinction between them?
At the particle level the classical electromagnetic field does not exist, it is interactions between photons and the rest of the particles that holds, and yes, there is a difference if a photon hits an electron or a proton for example. The study of scattering processes is explored by quantum field theory which is a meta level in calculations, based on the underlying quantum mechanical solutions of free particles. It leads to Feynman diagrams which allow calculations for many body problems.
Using the tools of quantum field theory one can show that the classical electromagnetic wave emerges from the underlying zillions of photons , a quantum mechanical superposition of zillions of photons.
There is no essential distinction between particles, of which the photon is one, except their masses, spins and other quantum numbers. The photon is one of the elementary particles in the standard model.