How is Regression different from Econometrics?
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.