Technically, all magistrates (magistratūs) had the power of veto, and this included the top levels all the way to the bottom: dictator, censor, consul, praetor, curule aedile, and finally quaestor. Any magistrate could veto the action of any other equal or lower magistrate (note that it only applied to magistrates: not to the Senate except in senatus consultum, not to the decisions of juries—in iudico—, and never to the decisions of military leaders in the field). If one magistrate vetoed the action of his equal, it was called par potestas ("equal power" ie nullification). If a magistrate acted against someone of inferior power status, it was called intercessio. The veto of a consul against a lower magistrate is very, very rare. In fact, I can't find a good example. The "veto" from non-tribunes was nearly always par potestas against a colleague of equal status. We hear of consuls doing it to their colleagues, and of praetors against other praetors; I can't summon any examples of veto by aediles or quaestors. I'm not sure about censors (they are a special case).

The concept of veto goes all the way back into the dim beginnings of the Republic system, and seems originally to have been intended to check the actions of any individual magistrate. It is intimately connected with the concept of dual or multiple magistracies (like there being two consuls). The veto power of the tribunes of the plebs comes later, but it is modeled on this slightly older concept. Tribunes were vested with potestas maior ("greater power"), which gave them the ability to invoke intercessio against any magistrate (except a dictator). Since they were technically not magistratūs, their own actions could not in turn be subject to intercessio from any magistrates. There were also certain rules about "vetoing a veto."

Therefore, the answer to your question is: the parameters and origins of the veto were ultimately the same for both consuls and tribunes, though technically a tribune's veto "out-ranked" a consul's.