Aberrations, or optical imperfections, exist because no lens is optically perfect. Almost all lenses show aberrations from the ‘perfect’ image.
Diffraction is an image softening effect caused by very narrow lens apertures. It happens because light tends to bend when it passes over the edge of an aperture, and with small apertures a larger proportion of the light is diffracted (scattered).
Diffraction is less of an issue with larger format cameras (cameras with physically larger sensors) because the lenses are correspondingly larger, as are the aperture mechanisms inside the lenses used to control exposure.