Difference between "doing art" and "making art"

Solution 1:

Doing art has a more pedestrian feel than making art.

Do verb 1.8 Learn or study; take as one's subject. ‘I'm doing English, German, and History’ - ODO

Doing art can mean undertaking a course of study in art, just as one can do Engineering, Maths or English.

In the context of your question, though, doing art is contrasted with making art, not 'doing' some other course of study. In this context, saying that one does art refers to the mechanics of the endeavour, whereas saying that one makes art focuses on more sublime aspects such as creativity, beauty and so on. This stems from the way do and make are used.

Do verb 1.1 Perform (a particular task) ‘I spent this afternoon doing Christmas Shopping.’ - ODO

Make verb 1 Form (something) by putting parts together or combining substances; create. ‘my grandmother made a dress for me’ - ODO