Both the Roman Republic and Roman Empire had periods of instability which ultimately caused their demise. Interestingly, each lasted for almost the same amount of time, so it would be difficult to ascertain one as more "successful" than the other. However, if one considers the extent of the Roman Empire and its continuation from Byzantium after the collapse of the Western Empire, then the Empire was ultimately more successful than the Republic.
The Roman Republic controlled most of the Italian Peninsula and soon spread into the Mediterranean, where it came into conflict with the Phoenicians, primarily over colonization and control of Sicily. Although the Romans won all three Punic Wars, the cost had been expensive and those who fought in the wars had been forced to abandon their homes and farms which were in disrepair. They sold their farms to a number of wealthy citizens and thereby lost their right to vote. The end result was a civil war and later a slave revolt led by Spartacus which ended with a number of generals raising private armies, among them Gaius Julius Caesar. Although the dying republic was ruled for a short time by the famous triumvirate; it soon collapsed with the death of Crassus and Julius Caesar became the first Roman Emperor.
The Empire itself lasted no longer than the Republic, but its territorial gains were considerably more extensive. Whereas the Republc had limited itself to the Italian Peninsula and the Mediterranean (which the Romans called mare nostrom: "our sea) the Empire extended to Britain in the West, and to the banks of the Rhine and Danube Rivers in the East. It encompassed vast numbers of people and thereby spread Roman culture throughout Western Europe. On this basis alone, one might argue that the Empire was more successful than the Republic. One should also consider that the vast extent of the Empire resulted in its division into Eastern and Western Empires; and that after the fall of the Western Empire in 476, the Eastern Empire continued and was quite successful until the fall of Constantinople to the Turks in 1453.