Art and Design both can be very similar to each other but contain different qualities. Art is a form of expressing ones feelings through a painting, drawing, music, etc. Typically there is no restrictions when creating a piece of art because it is something you do freely that creates an emotional power for the viewer. This allows for the artist to express themselves in any form of media, without needing to explain why they created it a certain way. The viewer can have their own perspective on the piece while others may differ.
On the other hand, Design has to do with rules and clear messages. When designing you have more planing involved than you would in just creating a drawing,etc. You have to be able to have a sense of what color combinations work well together and what objects you can use to catch the viewers attention. Designer are limited to several things because they receive a brief which is what is to be included in the design, restrictions. When designing, you need need to convey a clear message to the audience so that all receive the same one.
Many people can argue that art and design are both the same. Design is included to create art and art is included to create design, but in the professional fields it can rather differ. Take for example, a portrait painter and a graphic designer.