Spectrum is band of 7 colours while dispersion is splitting of light in to 7 colours
Splitting of light in different colours according to their wavelengths
Splitting of white light into 7 colours known as spectrum

Dispersion is the process of splitting of white lightSpectrum is the end result of dispersion. i.e. the colours of the rainbow
The phenomenon of splitting up of a white light into its constituent colours is called as dispersion. … When white light passes through a prism, it spreads out into a band of seven colours,VIBGYOR. This band of colours obtained is called spectrum.Dispersion of light results in the formation of a spectrum.