I understand after Julius Caesar's assassination, there was a power struggle which eventually led to his nephew taking power and claiming himself the first Emperor of Rome.
What I don't understand is what changes occurred within the political system of Rome itself between the two government types. [Edit:] Did the Senate never reconvene after Caesar's death? Did the Roman Empire retain any elements of a Republic afterwards or was it literally a large dictatorship at that point?
[Edit2:] If Julius Caesar is credited with claiming himself Emperor... and he died shortly after making that claim. Why didn't Rome just continue as usual?
Oh, goodness. There are books to be written about this. I mean actual books; hundreds of years of work.
The first thing to point out is that the notion of Republic vs. Empire is a modern construction. Velleius Paterculus, writing during the reign of Tiberius called the state a Res Publica. I'm not familiar with anyone before Tacitus definitively claiming the Republic had ended. Augustus never described himself in terms we Anglophones understand as "emperor" (he did use the word "imperator" but that did not refer to a system of government).
The second thing is that almost no formal changes happened under Augustus. Magistrates kept being elected, the Senate continued to meet, laws continued to get passed. The real change was informal - the Senate has a problem with Germans; let's go ask Augustus what he thinks we should do. Oh, Augustus has this stepson that's really handy with Germans - why not give him an army? It also helps that Augustus was probably the richest guy around.
I like to use the opening scene of The Godfather as an illustration of how Augustus ruled Rome. It's all personal connections - not legal authority or military force.
Also, I'm writing in an airport; I don't have my books. I can update in a couple of days if needed.
[I'm going to pull from an earlier answer I wrote here,] (http://www.reddit.com/r/AskHistorians/comments/1kgclt/after_julius_ceasar_how_many_legitimate_chances/) and expand on it:
The Roman imperial system was initially built by Augustus on top of the Republic, instead of replacing it. "The honors race" amongst leading families, the civil bureaucracy, the religious and legal systems were all in place, even the Senate remained in effect, although it's informal power (it was never explicit under the Republic) was no longer the dominant force it had been. Instead, the Emperor served as a final authority to govern political conflicts at home and prevent the violent factional conflict that was typical of the late republic. This system worked under good Emperors, and could survive the dysfunction of bad ones (for a while.) The imperial system also effectively reworked the Republic's institutions so that they would be effective in raising, supplying, fighting and retiring the legions in the far flung provinces which was always difficult under the Republic.
However, this fact did not sit well with the upper class Romans which comprised the Senate. The Senatorial class in Rome was the primary source of top political and military talent in the Republic, and the Empire needed this pool of talent for its administrators and soldiers. Over time, the army, and the equestrian class became alternative sources of educated and trained personnel. Senators resented the rising equestrians and the loss of power of the Senate itself. Nostalgia or agitation for the removal of the Imperial system from the body of the Republic was therefore common.
However, the Senatorial aristocrats could never really turn back the clock. Patronage and faction was a way of life for them, and being unified could never last. A patron who could assemble a strong enough faction to topple the Emperor would himself be a candidate for the imperial role and a threat to the other Senators.
Edit: This is loosely sourced from the ideas in Antony Everitt's biographies of Cicero, Augustus and Hadrian.
The key difference between the early Empire and late Republic was that the Emperor at the top of the system acted as a check on factional conflict that was endemic to the late Republic. The Republic had fought major civil wars with alarming frequency between 91-30BC. The Imperial system kept a lid on factionalism because it promoted, thru the same official ranks the Republic used, only men who were loyal to the Emperor, or part of his network of clients. He was therefore in a position to quash conflicts between high ranking senators in a way that could have been resolved thru armed conflict at the end of the Republic. The Emperor also resolved the longstanding governance issue in the late Republic whereby officials blocked each other from resolving important problems in order to prevent a rival from increasing his power and prestige.
Eventually however, the Imperial system began to erode the prestige, wealth and power of the Senatorial class. Over many generations the Republican institutions withered and were replaced with a system in which political conflicts could only be resolved by deposing the Emperor, leading to renewed instability and frequent civil war.
If HBO's Rome is being historically accurate, the reason the Republic didn't return back to where it was before; may have been due to politicians trying to keep their seats after Caesar's murder (by keeping his will intact, which instated Augustus as heir) vs their appointments being annulled (corruption) which would have required new elections.
Huh... Not accurate