What is the difference between a normale mode of the EM field and a photon ?
Yes, photons are normal modes of the electromagnetic field. In classical E&M we can expand the electromagnetic field into spatial normal modes. Classically the EM field can be in a superposition of different normal modes but energy is fixed.
Quantum mechanically we can still expand the EM field into normal modes. The difference is that the possible states of energy are quantized. This means the amplitude of the EM field is quantized. However, quantum mechanically the field can be in a superposition of energy states. This means the expected value of the quantum mechanical energy in the EM field can take on a continuous range of values, but the expected value of the energy does not uniquely determine the exact EM field superposition.
So in short, yes, a photon is a single quantum excitation of a normal mode of the quantum electromagnetic field. This is the best definition of a photon that I know.
Modes are ways of looking at methods of radiating relatively low frequency electromagnetic fields using what are classical electromagnetism where the concept of individual photons is not useful and the radiation is expressed in terms of voltages and currents.
The transfer from classical to quantum techniques starts around the millimetre wave to infra red regions of the spectrum, except for certain special cases in either direction where quantum processes (21 cm hydrogen spectral line) or techniques like optical TE10 waveguides are involved.
Strictly speaking, a photon is a number state or Fock state of a quantum field. This is the case if you can express the Hamiltonian operator in terms of a definite particle number operator. Below is an illustration of the Wigner function (a sort of phase space distribution) of a single photon state |1> for a harmonic oscillator potential (source: Wikipedia).