Manufacturing implies an artificial process for creating something (usually a physical object). Besides simple examples like cars, airplanes, you can also manufacture evidence in a court of law. The implication is that you made it up, or falsified the evidence. In contrast, to produce evidence means to present it to the court.
Manufacturing also implies a more mechanical, and possibly dehumanizing process than production. To say you produced a work of art implies you have exercised artistry and creativity, whereas to say you manufactured a piece of art would mean you either copied it or otherwise followed some rigid formula.
For instance, some people say that President Obama's speeches felt manufactured. What they mean is that his speeches did not feel genuine, or that it represents his own ideas. Something that is manufactured is the product of a machine, whether it's a physical machine, some algorithm, or some organization.