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That is a great question, and one which also concerned Einstein. He was never satisfied that our current understanding of photons (and other quantum mechanical particle/waves) is entirely satisfactory. The following is a comment Einstein made near the end of his 1917 paper on the Quantum Theory of Radiation (which is the paper in which he introduced his discovery of stimulated emission — this discovery is essential to the operation of a laser, although it took another 50 years for that to happen).
“The weakness of the theory lies on the one hand in the fact that it does not get us any closer to making the connection with the wave theory; on the other, that it leaves the duration and direction of the elementary processes to ‘chance’. Nevertheless I am fully confident that the approach chosen here is a reliable one.”
In other words, photons do very definitely behave like particles in many respects. In fact they behave like highly relativistic particles, as the relationship between the energy, momentum and pressure of light is the same as that of classical particles which are moving at velocities near the speed of light. On the other hand, light also has very easily measurable wave-like properties such as diffraction. There are also numerous fascinating experiments which demonstrate that a single photon is able to pass through two spatially separated holes (or slits) while a classical particle could obviously only go through one hole or the other.
I hope that helps…if you are still confused…you are not alone.
I will read this in the evening, but my impression was always that it’s a problem of metaphor. We cannot relate to the experience of a photon and its interactions, thus we cannot feel comfortable calling it what it is.