What is an econometrician? According to Wikipedia, Econometrics is the application of mathematics, statistical methods, and computer science to economic data and is described as the branch of economics that aims to give empirical content to economic relations. It sifts through mountains of data to extract simple relationships. Generally for positions like Analytic Consultant, Data Scientist, Statistician, Quantitative Analyst, background in maths, stats or econometrics is required. What separates econometricians in terms of what they learn in school and how they apply acquired knowledge in work? Or do these fields overlap so much that there is not much to distinguish?
An interesting question that leads to a debate among econometricians. Some consider that 

Econometrics is just statistics applied to economic problems


  Econometrics is just statistics applied to economic problems—nothing
  more and nothing less. We should probably call it “statistical
  economics,” but I guess people feel that the term “econometrics” has a
  better ring to it. The only cost of using the term “econometrics” is
  that we are sometimes fooled into thinking that we work on a distinct
  discipline, separate from statistics. This is not true. 


From John Stachurski at ANU (see his notes p.108 in Econometric Theory freely downloadable)

Others consider that

Eonometrics by no means the same as economic statistics

This quote is from Ragnar Frisch (one of the founders of the Econometric Society) and his Editor's Note on why the Society has decided to establish its own journal, Econometrica.

The key idea is that economic theory is crucial to 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.

Another difference is, as economists, we are biased towards establishing causal relationships. So, this explains our focus on endogeneity issues and strategies of identification.