The EdX course seems to focus more on LP theory and for example also discusses the simplex solution algorithm. That's helpful for understanding what a solver actually does you tell it to optimise your model. A downside is that they only seem to focus on continuous problems.
The Coursera course appears to be more of a tour d'horizon of mathematical programming approaches, discussing many applications and models. This is very helpful to get a broad understanding of what sorts of problems can be solved using mathematical programming models, but it does not really help you understand how these models are solved. Of course, the second and third courses in the OR specialisation will remedy this knowledge gap somewhat, but then you have to commit to more than just a single course.
I think you will learn more from the three course specialisation on Coursera, but you might also want to pick up a standard textbook on linear optimization as a reference work/aid during the course, like Bertsimas' and Tsitsiklis' book. Then you are sure to get a solid understanding of the basic mathematical programming tools used in the OR field.
http://www.amazon.com/Introduction-Linear-Optimization-Scientific-Computation/dp/1886529191 Operative Research with geometrical view of linear programming