What is the relationship between the energy differences of the orbits in an atom and the light emitted by the atom?
One of the postulates of Bohr's atom is -
When an electron makes a transition from a level of higher energy
Thus energy levels and transitions between these levels determine what frequency of light is emitted when an electron returns from a higher energy orbit to a lower energy orbit.
This is what gives rise to the discrete lines in the emission line spectra of a particular element. Where each line corresponds to a particular frequency which in turn corresponds to a particular transition between orbits in the atom.