How are raleigh scattering and compton scattering different?

Rayleigh scattering is an optical phenomena where blue light is scattered and Compton scattering is the dispersion of radiation with a greater wavelength after the collision between a foton and an electron.

Explanation:

The Compton scattering is related to electromagnetic radiation that incides on electrons, this radiation after hitting an electron is dispersed with a wavelength greater than the original. In this phenomenon the conservation of the linear momentum and the conservation of energy are applied to analyze the clash between the photon and the electron.

And the rayleigh scattering is the dispersion of light due to air molecules, this dispersion is more effective for small wavelengths (the blue part of the spectrum).

So the difference is that the Compton scattering has to do with collisions between particles, photons and electrons, and the rayleigh scattering is an optical phenomenon of white light scattering, where mainly the blue part of the spectrum is scattered and is used to explain, among other phenomena, the blue color of the sky.