In a drawing, what is the difference between defining an edge with a dark line versus defining an edge with a change in value?
The two methods of defining edges you have described in your question indicate two separate types of drawings, contour drawing and chiaroscuro drawing.
A contour drawing is one that is characterized by line quality. In this type of drawing the artist uses a line to define an edge. The eNotes/Wikipedia page on Contour drawing defines it as
When an artist uses this type of drawing method, she must show sensitivity to her mark making and line quality. Some guidelines to govern line quality in a contour drawing include making your lines darker and/or thicker at the bottom of each object to simulate a shadow, making your lines darker where one object overlaps another, and making lines get lighter and thinner as the edges they define get further away.
The word chiaroscuro comes from the Italian word for light and dark. In a chiaroscuro drawing, an artist attempts to draw the reflections and shadows caused by the light as it hits the objects in a still life. The edges of objects are no longer defined by a line but by an area of dark value next to an area of lighter value. Because this type of drawing attempts to capture the way we actually see objects with light hitting them, it is considered a more realistic process. By this I mean that it will yield a more realistic picture, if done with skill.
According to the eNotes/Wikipedia page on the subject, a chiaroscuro drawing,
is characterized by strong contrasts between light and dark, usually bold contrasts affecting a whole composition. It is also a technical term used by artists and art historians for using contrasts of light to achieve a sense of volume in modeling three-dimensional objects such as the human body.
I usually have my students begin their stilllifes by drawing the contours of each object. However, when they are sure that they have drawn the objects the right size and in the right place, their job is to get rid of all the lines by shading with the appropriate values on either side of the line. It is the mark of an inexperienced artist to combine the two techniques of contour and chiaroscuro. A chiaroscuro drawing that still shows evidence of contour lines appears to be unfinished and, consequently, of lesser quality.