How is Regression different from Econometrics? I am an undergraduate student and am planning to take a course in Regression Analysis next semester whose description is 

"The analysis of continuous response data. The focus is on linear and multiple regression with theoretical and practical training in statistical modeling. A hands‐on, applied course where students will become proficient using computer software to analyze data drawn from a variety of fields, and will learn what assumptions underlie the models, how to test whether the data meet the assumptions, and what can be done when the assumptions are not met. "

I am signing up for it because the university I attend currently does not offer a course in Econometrics and Regression is the closest thing to Econometrics that is offered. I want to understand Econometrics at the level of Stock and Watson's Introduction to Econometrics and/or Jeffrey Woolridge's Introductory Econometrics textbook. 

Can someone please explain what the differences might be between the Regression course mentioned above and a course that would be taught out of the Econometrics textbooks mentioned above? If in a hypothetical world, I could take a course next semester supplementing the Regression course to patch up my understanding of Econometrics, what book/ source would you recommend that I read? Any advice is greatly appreciated! 

Aside: I want to get acquainted with the world of economic research and eventually pursue a doctoral degree in Economics.
To put it simply regression modeling is a technique, a tool, used by econometricians. Regression deals with dependence amongst variables within a model. But it cannot always imply causation.

As economists, we are biased towards establishing causal relationships. So, this explains our focus on endogeneity issues and strategies of identification to establish causality.

In other words, as Ragnar Frisch (one of the founders of the Econometric Society) would say


  Econometrics by no means the same as economic statistics


See also the answers to What is an econometrician? where the key idea is that economic theory is crucial to econometrics and understand measurement. Neither theory nor measurement on its own is sufficient to further our understanding of economic phenomena. We need both and measurement without theory is unlikely to provide a satisfactory explanation of the way economic forces interact with each other. 

In summary, regression is super useful but not sufficient to make causal inference and to understand economic phenomena. 

Stock and Watson's Introduction to Econometrics and Jeffrey Woolridge's Introductory Econometrics are both excellent textbooks.