I will give some background on the current historiography of the kings first. Due to the loss of contemporaneous records after Brenuss' sack of Rome circa 390 BCE, the Romans were given carte blanche to rewrite their history, which they were not too shy to abuse, great examples of such being the thematic names of rulers, "hostillius" for his hostilities, "Pompillius" for his pompes (Greek for processions, hence pomp, as in "pomp and circumstance), and 1 upping Athens by pinning the Roman Republics founding 1 year before theirs. So in this clearly idealized framework, there was correspondingly an idealization of absolute power that the Kings held, absolute imperium, which in reality probably had many more informal checks than we are explicitly aware of; this is hinted at by the existence of the early Roman Senate, which means collection of old (or in better terms, "wise") men, and the word Senate and senile come from the same roote for old/wise. This Senate had 1 function outside of being advisory, it chose kings in interregnums where each senator would have kingly power for 5 days to attempt to find and appoint a new king which would need to be agreed upon by the rest of the Senate until the new king could be appointed. But in effect, the kings had ultimate political and religious authority in Rome.

The Empire, being much more accurately recorded, also has much more niche when it comes to the emperors power. Again, the Emperor had supreme imperium, being the Imperator after all, but the way in which it was exercised was vastly different throughout the course of the empire.

I will begin to introduce some modern political vocabulary here that should be familiar, in particular the legislative, the power to make law, and executive, the power to enforce law. In the kingship there was an absolute concentration of power among the executive, and even though the king could "legislate," his power as such would make such a process quite ad-hoc and arbitrary, insofar as a king may impose his whim and effectively fully devalue any semblance of legislative procedure. By contrast, the beginning of the Principate was a revival of the executive which has been repressed by the Republican Senate about 5 centuries, but this was not another sudden flip from one pole to another, but a gradual transfer of power. Then, when the Emperor had centralized all meaningful power with no necessity to pay lip service to the Senate, the dominate came into full effect.

Augustus titled himself "princeps" after the battle of Actium and restored the veil of the Roman republic to appease the fears of the patriciate of a new king, which were still much alive. This optical posturing is precisely what kept the legislature, and by extension the patriciate, alive in Rome a couple more centuries, and the continued importance of both institutions is what characterized the Principate. Some senators of augustus even attempted to pejoratively use "dominus," or my lord (the title slaves would address their masters by), when directly addressing Augustus, which Augustus flat out refused to accept to keep the veil of the republic alive. During this time there was back and forth political interplay and intrigue being exercised by both the Princeps and the Senate, however the realities of the institutions after some time had been lost on nobody and the Senate usually relegated itself to a pressuring body toward the Emperor or the Emperors biggest fans club, or even fulfilling the duties of the ancient regal senate of interregnum from time to time. Also after the death of Lepidus, Augustus subsumed the title of pontifex maximus; along with divi Filius, this made him the supreme and ultimate religious authority in Rome which stuck with the Emporer's post until the end of the Empire (or the political muddling of western Christianity depending on how you see it but no way in hell am I unpacking that can of worms here).

Due to the political innovation of bigger army diplomacy getting too out of hand during the empire, hence many a civil war, paying lip service and playing along in the little Roman Senatorial game was just seen as too costly in the fallout of the Third Century Crisis, so Diocletian styled himself as "Dominus" to make it absolutely clear that he, his co-augustus, and the associated Caesar's (the leaders of the tetrarchy) were equals, everyone else to them slaves. This was likely extremely similar to the power of the original regal post in Rome.

This was the last great political, but not cultural innovation to to centralize power in the Emporer, who effectively became but a hegemonic warlord in a country that needed warlords to keep it's borders in tact, but also had them frequently engage in contest with one another and the Emporer, one of the biggest detriments to such despotism was in fact that it was so sought after. But at this point when institutions, traditions, and morality are not even given a semblance of respect, the Empire became a supreme warlord over squabbling ones, and whatever military camp the Emporer was at became the capital, his inner circle becoming the core bureaucracy.

Bureaucracy is probably the biggest difference between the dominate and Regal Rome, since the kings had no needed for such divided administration to effectively rule their realm, but a continent wide one required almost innumerable political abstractions, which became a problem because basically there is no difference between corruption and the normal way of doing things in Rome (see patronage) and in this the inefficiencies when combined with natural fluctuation could diminish the taxable base to near 0 for some Emporer's.

This was a dangerous path to decentralization and faux Emporer's being subordinate to foreign warlords, and while Diocletian's Imperium would have been very similar, if not nearly identical in it's political power to that of Romulus', Romulus would look down on his nose at the sniveling, sorry excuse for an Emporer that Romulus Augustulus was.