In my personal opinion, the difference between these styles comes from how the painter perceives the subject. As humans, we all see the world differently. Some people concentrate on the big picture and overall composition of an entire scene; others are drawn to the small details that might be hidden to other observers.
"Linear" painters concentrate on the lines that certain features form. In your right example, the hair flows in front of the shoulders and the hairline from the very tip of the forehead is continued in the slope of the neck. Other examples include representing movement by following bigger lines through the body, like this example from Design Host:
If you turn the linear style up to the extreme, you can reduce entire bodies or objects to simple geometric figures, like this other example from Design Host:
"Painterly" painters (or "sculptural", as Joachim calls it more correctly), on the other hand, concentrate on the 3-dimensional planes and areas instead of lines. The entire hair on the left image is composed of 3 planes: the majority in the center, a raised plane from the forehead to the front ot the neck, and a deeper layer from the very back of her head to her back. Other very common examples segment a face into its planes, like this one from Nat Studies Art!:
A "sculptural" picture usually portrays the subject as a 3D object, giving it depth by emphasizing the structure. To stay with the elephant example, here are some "sculptural" ones from iStock:
The thing that might be confusing is that using a lot of lines to paint a picture doesn't make it a "linear" picture. Hatching and other shading techniques are common and don't define the art style.
I would still call this example from Do It Before Me a "sculptural" drawing, although the depth perception is created by lines:
However, the example below from Cherish LoveArt is (in my opinion) a "linear" drawing because it emphasizes the lines and textures of an elephant's skin: