Scalar Electrodynamics or QED?

In scalar electrodynamics you still have the four-vector potential $A_{mu}$ but no spinors, that is no Dirac fermions. Spinors are solutions to the Dirac equation (QED) that take their name from the fact that they have spin (=1/2). In scalar electrodynamics, that is, the fields do not have spin - i.e. they have spin $0$. The field is a scalar.

Say you have experimental data of collisions between electrons, mediated by electromagnetism (as opposed, for example, by the weak force).

You can calculate the theoretical predictions both with QED and with scalar electrodynamics.

They might both agree with the data, for instance, at low energies. Which tells you that the spin of the electron does not play a crucial role in that energy régime.

But only QED will agree with the data across the whole spectrum, since electrons do have spin and QED is the best theory we currently have to describe them.