Going by what I found online, it looks like
comenzar really are pretty interchangeable, with
empezar being more common in Spain and
comenzar more common in Latin America. Both mean "to start", but
comenzar is generally translated as "to begin" because (in Spain at least) it is seen as a bit more formal/literary.
Iniciar is more formal/technical than either, with "to initiate" being a decent translation.