There were many important continutities between the Middle Ages and the Renaissance. Let us look at two of them. First, there were important continuities in political organization. Much of Italy, for example, continued to be ruled by popes or by lords in a fairly feudal system. The same is true for other parts of Europe where feudalism was even stronger than in Italy. Second, there was a great deal of continuity in religious belief. The Renaissance thinkers were known for humanism, but were not, for the most part, shedding traditional beliefs. This was still a very religious age.
That said, there were important changes going on. The rise of humanism and the beginnings of rejection of authority are important changes that came with the Renaissance. This idea that people should study the world as it is and try to determine how it works in an objective way, using scientific reasoning, was something that was clearly new.
First, we can look at religion. Most of the Latin west remained Christian in both periods. In the Middle Ages, though, the Roman Catholic Church completely dominated the religious environment, challenged only by a few minor movements such as the Lollards and the Albigensians (Cathars). The Renaissance marked the period of the Protestant Reformation, in which western Christianity was divided into multiple denominations.
Although educated people in both periods knew Latin, the vernacular became increasingly important in the Renaissance. European vernaculars evolved rapidly and by the Renaissance took the form of the modern languages we still use.
The late Middle Ages were a period of exploration and technological innovation, but the pace of these developments accelerated in the Renaissance. One major difference was the Renaissance involvement of Europe in the Americas and the rapid growth of colonialism.
Although there were many political continuities between the two periods, the Renaissance marked the consolidation of monarchical power in France, Spain, and England.