How Did the Roman Republic Differ from the Roman Empire?

The Roman Republic was a plutocratic oligarchy. As is generally true of republics, it was a mixed constitution with a monarchic component, the Consuls, and aristocratic component, the Senate, and a democratic component, the Assemblies of the Plebes. The plebes had their representatives, the Tribunes of the Plebes who could veto proposals of the Senate. It was the Assemblies of the Plebes that actually passed laws.
While there were elections, they were heavily weighted in favor of the upper classes. Voting was done by centuries, and the centuries of the upper classes contained far fewer people than the centuries of the lower classes. (Think of the U.S. Senate in which Wyoming with 600,000 residents gets two Senators while California with 49,000,000 residents gets two Senators. In which state does the average citizen have more power?)
To gain authority in the Roman Republic, an ambitious Roman first had to serve for ten years in the Roman legions. He could then embark on the “Cursus Honorum” a series of elective offices-Quaestor, Aedile, Praetor, and the highest office, Consul. After serving in one office for the term of a year, the politician was eligible to serve in the Roman Senate, and when not serving in office most politicians would be serving as Senators. Nobody attained the highest office in ancient Rome without significant experience in both military and civic affairs. (Unlike the American Republic where a person can be elected President with absolutely no experience in military or civic affairs.)
One of the ideological pillars of the Roman Republic was a distaste for monarchical rule. When Scipio Africanus won the Battle of Baecula in 207 B.C. he freed the Spanish prisoners who had been fighting for Carthage. They haled him as the Roman King. He told them “Under no circumstances are you to call me “King.” We Romans do not have a king and the very word is distasteful to us.” Indeed, being accused of wanting to be a king proved fatal to some prominent Romans such as Marcus Manlius Capitolinus, the hero of the Capitoline siege, and to the reformer Tiberius Gracchus.
So what went wrong with a system that had endured for nearly five centuries? Traditionally, a citizen had to own property to be admitted to the Roman legions. In 105 B.C. the Roman legions suffered a devastating catastrophe at the hands of the Germanic Teutons and Cimbri at the Battle of Arausio. It was an even worse defeat than the Battle of Cannae, with 80,000 Roman legionaries and 40,000 servants and camp followers slaughtered. The Roman politician Gaius Marius was elected Consul and to deal with the Germanic threat, he made reforms in the military allowing any citizen, whether they owned property or not, to join the legions. This measure paid off in the short term, when in 102 B.C. Marius’s legions soundly defeated the Teutons and Cimbri at the Battle of Aquae Sextae, ending the Germanic threat. In the long run, it destabilized the Republic because the state lost control of the militaries, and the legionaries came to be beholden to their generals rather than the state. There followed a series of civil wars with Romans fighting Romans: Marius versus Sulla, Pompey and Metellus versus Sertorius, Catalina versus the Roman government under Cicero, Pompey versus Caesar, Octavian versus Marc Antony, Octavian and Marc Antony versus the assassins of Caesar, and finally, Octavian versus Marc Antony and Cleopatra.
Some of these civil wars were followed by proscriptions in which wealthy Romans determined to be enemies of the faction in power were deprived of life and property. If you found yourself on the proscription list, the best thing to do was leave the city as quickly as possible. Your slaves would be rewarded for turning you in. Between civil wars and proscriptions, the Roman aristocratic class was so weakened that when Octavian took power in 31 B.C., there was no effective opposition. The lower classes were thoroughly sick of the civil strife and disorder that they welcomed the peace and tranquility that Octavian offered them. They were content with their bread and circuses.
With the ascension of Octavian (Augustus) to power, Rome became a monarchy. There were still elective offices but those who were elected were endorsed by the monarch, who was now called Caesar. The Senate served at his pleasure, and he appointed the governors to the provinces. He determined who served in public offices, and who served as military officers. In the case of Augustus, the military leaders were generally his close relatives such as Tiberius and Germanicus.
The problem with the monarchy was that there was no clear method of succession. This tended to cause instability, and could result in civil war. The monarchy worked best when the Caesar chose his successor on the basis of merit, which was the case during the second century A.D. with the five “Good Emperor”, Nerva, Trajan, Hadrian, Antoninus Pius and Marcus Aurelius. Once this practice was abandoned, with power passed to a son might not be competent, as with Commodus, the system broke down. Beginning in the third century A.D. the Roman government was never stable for very long, and assassinations and power struggles became the norm rather than the exception.