Smasif is right. well explained...just one part where I would like to add some more information...drawing does have commercial applications but not the drawing you do in our notebook when u are bored in office
. Drawing is the root of whatever final product you make. Lets say in case of web-designing most of us would draw on our sketchbook to visualize how the final site may look like. Now imagine the movie Lord Of The Rings....the directors didnt just come up with those breath-taking scenes...they are usually visualized or drawn by the movie's concept artists and writers. Once drawn and approved they are tweaked, perfected so that they can be used commercially as concept artworks or supporting materials. Now some ppl call these illustrations because as smasif already said...illustration is a refined work of a concept drawing or sketch which now has more depth and story in it and that is more presentable to general audience. Depends from situation to situation...
Design cannot be compared with drawing in any circumstances (anybody can correct me if i am wrong). The reason is design is an arrangement of various elements that are put together in one place in a nice and presentable manner. Lets bring the Lord Of The Rings example here...we can use that nice artwork of that movie on a DVD cover...but just using the artwork on a DVD cover might not be presentable enough for getting people to buy the dvd so we would add some relevant information like movie name, some fancy or catchy phrase, etc...maybe some design elements surrounding the artwork as well...so you can see we are using several elements and composing something that might have more meaning for intended purpose...in this case the purpose is marketing the movie.