How did Renaissance artwork look different from artwork during the Middle Ages?

Because the field of artwork is broad, I will focus on painting. There are many ways in which medieval painting differed from that of the Renaissance, which is what makes the latter period so important as an era of innovation.
First, nearly all medieval painting is religious in its subject matter. A great deal of Renaissance painting is, too, but more liberties are taken in the portrayal of the icons. Renaissance painters individualized the figures, allowed them to emote, showed the natural movement of the body, and often depicted subjects according to classical standards of beauty. Medieval iconography is less inspired and quite redundant. 
Moreover—and this leads to the second major difference—there is no depiction of perspective in medieval painting. The figures appear flat and two-dimensional. Vanishing point perspective developed during the Renaissance. With this technique, a set of parallel lines are perpendicular to a picture plane, drawing one's eye to a single point in space, then outward. This allowed for the illusion of depth on a flat surface.
The depiction of children is another very important difference. Both medieval and Renaissance painters were interested in the depiction of Madonna and Son. Based on the images, however, one thinks that medieval painters had no concept of what children actually looked like. They are little adults: elongated and earnest in their expressions. Renaissance artists painted babies as they are in real life: chubby, playful, and fidgety.