To simply something is to make something simpler. To dumb something down (the process known as "dumbing down," the past tense being "dumbed down") is to make something simpler by making it less intelligent.
Example: A frequent theme when discussing education in Western countries is whether the education curriculum has been dumbed down in favor of "trendy" and flashy subjects. It is argued that this leaves students of the current generation less prepared for academic life and employment once out of high school.
To simplify has a positive connotation. Dumbing down always has a negative connotation. "Dumbing down" or "dumbed down" is never intended as a compliment.