I am currently working on an application which deals with the restoration of fluorescent beads examined using widefield microscopy. There are two major sources of noises that are being encountered in the image: photon shot noise and salt and pepper noise. I have read in a paper that, both of them can be filtered using a median filter. So, is the essential difference between them only their sources?
photon shot noise follows a Poisson Distribution, which is essentially a shifted Normal Distribution. It is an electrical noise usually caused by charge surges. Salt and Pepper Noise is more due to digital conversion, and is usually modelled using a Fat-tailed distribution (which is essentially a shifted Normal Distribution). What to note is for any unaltered image, the order of magnitude of each type of noise, with photon shot noise usually much lower than salt and pepper noise (unless specifically created)
True both noise can be treated using median filter, and often people use salt and pepper noise to represent shot noise.
Photon shot noise arises from the random statistical fluctuations of photons colliding with a detector. Often, the photon source creates a Poisson distribution which means the probability of each photon arriving at a given time is independent of the other photons. Here is an application note for measuring shot noise with low-noise photodetectors which explains shot noise in more detail.
Salt and Pepper noise is classified by the resemblance of white and black pixels in an image. There are many sources that could cause this noise, but as GameOfThrows mentioned, digital conversion error is a common source.