How can the difference in the brightness of spectral lines be explained

The brightness of a line depends on the number of electrons making that transition.

Explanation:

In atomic spectra, some electronic transitions are “forbidden” by the rules of quantum mechanics. That is, some transitions are less probable than others.

The brightness of a given line depends on the number of electrons making that transition.

Allowed transitions will give the brightest lines. The brightness of the other lines will vary depending on how likely the transitions are.