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The key difference between the Photoelectric effect and the Thompson effect are that the Photoelectric effect is based on kinetic energy of photons and the Compton effect is based on light scattering and momentum.

The difference between the Photoelectric Effect and Compton Effect are the nature of the interaction of light with electrons. In the photoelectric effect kinetic energy is the main player due to the fact that a photon is a particle and a wave. The kinetic energy of a photon ejects an electron. In the Compton Effect, the important aspect is that it is a light scattering effect and depends on the angle of the photon and the momentum.

The Photoelectric effect deals with ejecting electrons from metals exposed to UV. There is an energy threshold you have to reach before electrons are ejected. The Photoelectric effect has to do with kinetic energy and the wave-particle duality. The photon is like a big truck hitting a little car on the highway. If the truck has enough kinetic energy it will put the car in the ditch. But it has to be high enough energy.

Light scattering is a more difficult concept to understand. The Compton Effect is the decrease in energy when x-rays are scattered off of electrons. Before Compton came along everyone thought light hitting and electron should result in scattered light of many wavelengths (classical physics not quantum). Since we know that E=hc/l, when the photon hits its energy decreases and wavelength increases. So if you start off with an x-ray you might end up with a UV photon after the energy gets taken by the electron. Because the Compton Effect deals with scattering, the angle is very important. Momentum is important as well. Instead of ejecting an electron, the Compton effect acts to increase the momentum. It gives the electron a little push so that it can go faster. It would be like getting hit by a truck but not going off the road and it would depend on which angle you were hit.

Here is my added intuition. At UV wavelengths you only have the photoelectric effect. When you get into the x-ray regime the Compton Effect kicks in. One site (Wiki) said that you get both processes happing at x-ray energies.

Reference:
P.W. Atkins Physical Chemistry 4th Edition

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