The above answer downplays the difference between the two eras; perhaps more than is accurate. There were substantial and meaningful differences between the two periods.

Learning and education in the Middle Ages consisted of Scholasticism, the doctrine that authority should be accepted without question, and that experimentation was dangerous, as it might lead one to sin. The two unquestioned authorities were the Bible and Aristotle, as scholars believed that the two were consistent. Education was closely affiliated with the church, and consisted primarily of the Triviam ("three roads", hence our word "trivia) and the Quadrivium ("four roads.") Those who disagreed with either might well find themselves condemned for heresy.

With the fall of Constantinople in 1453, the learning of the East long lost to the West was brought to Western Europe by scholars who abandoned that city. Much of this learning consisted of the ancient classics. The end result was the rebirth (in French, "Renaissance") of classical and secular learning. The literature and art of the Renaissance period often reflects Classical themes, such as nudity and similarity to Greek Gods even in religious works (e.g. Michaelangelo's "David.") Art of the period is often quite realistic, reflects emotion, and has perception; art of the Medieval period, in which human beings were considered corrupt and unworthy, is often two-dimensional, and reflects no emotion or passion. A prime example of medieval art is the Bayeux Tapestry, in which people appear almost as cartoon characters. 

The attention to human beings, noted in the answer above, was the direct result of the renewed interest in classical learning. This so-called "humanism," the belief that human beings are God's greatest creation, and therefore have some worth, led not only to the realism of Renaissance Art, but also to the development of an entirely new discipline, the so-called "humanities." Since it was believed that learning liberated one, the humanities became known as the "liberal arts."  

Perhaps the greatest change was the acceptance of challenge to authority by experimentation. This change was painfully slow: Copernicus, who challenged Aristotle's ideas of a geocentric universe, only published his findings at the point of his own death; and Galileo, who similarly challenged Aristotle in his Starry Messenger was forced to recant under threat of torture. Eventually, however, experimentation became the very nature and essence of scientific inquiry.

This change did not happen suddenly or even deliberately. It is perhaps the slowness of change which leads some to conclude that the differences were not that great.

Modern scholars argue that the Renaissance did not differ from the Middle Ages as much as was once thought.  However, there were still important differences between the two.  Some ways in which the two differed include:

• The society of the Renaissance was more of an urban society as cities were growing and becoming economic and cultural centers.
• The Renaissance represented a time of recovery.  It was immediately preceded by things like the Black Death which had devastated society.
• It was a time in which there was more awareness of the legacy of classical times.  
• It was a time when more attention was paid to individual human beings and their abilities and potential.

This is not to say that the Middle Ages did not have any of these characteristics.  It is more a matter of degree.  The Renaissance was a time in which these things were much more prevalent than they had been in the Middle Ages.