Read Judea Pearl's forthcoming Econometric Theory paper (2014). A regression model generally summarizes "see-type" relationships as found in the data, whether they be observational or experimental data. An econometric model generally looks to summarize "do- type" relations. It will agree with the see-type relationship in experimental data; but not necessarily with the see type relationship found in observational data. An econometric model asks for an answer to the question: What will Y be if I change X to be X + DX? Note with observational data a regression model may give us parameter estimates that do not correspond to the partial of Y with respect to X, due to omitted variables (what Pearl calls backdoor path variables). Read as well Pearl's 2009 book causality.

David's answer is interesting. (So you are looking at the purpose?) Question: Aren't omitted variables a problem in econometrics as well? - I use prediction for establishment sample survey work, which someone also found some use for in analytical chemistry, and I often think of the regression I use as very simple econometrics - without lags. In Ken Brewer's 2002 book on combining design-based and model-based survey inference, he cautions against using more than one regressor unless you are sure it helps. I would caution against overparameterizing a model, but i suppose the application and the size and quality of the data sets available determine justification of model parameters, so I suppose that too is related to purpose, though I think David's comments show more insight.

Something I have found interesting between say a regression analysis in a statistics class and a regression analysis in an econometrics class is the statistician cares that statistical theory is upheld. The econometrics analysis tries to "prove" economic theory.

If a conservative and liberal economist analyse the same dataset, they will come up with different models. If a conservative and liberal statistician analyse the same data set, the models will be very similar, if not the same.

Some of the reasons the economists will have different models is because of their "data cleansing" techniques. If you remove "outliers" based upon your model, you will remove data that contradicts your point of view. If you remove various terms from your model, based not upon the data but upon your theoretical beliefs, you will change the model completely.

I have always understood econometric models to be a subset of regression models. They are distinct in that they are (generally) based on economic theory rather than being purely exploratory, and there is a greater emphasis placed on model selection and forecasting based on parameter estimates. In academia, there tends to also be a distinction between pure econometricians (who adhere to orthodoxy like pure statisticians) and applied econometricians (mostly economists). See Peter Kennedy's (2001) enlightening article "Sinning in the Basement" on this distinction.

I agree with Derek and Andrew that the econometric models are a subset to regression models in which the economists tries to fit the model to some economic theory (which the statistician may not do - removing the "outliers").

I just revisited this question, and found it interesting to ponder it more thoroughly (including "data cleaning"), and the other answers given, which are quite helpful. 

The econometrician, as is the case for all other social and physical scientists, does not rely solely upon statistics. That field is a tool to be used by others in conjunction with theory.  Statisticians (the field from which I retired) may sometimes not be good problem solvers, because they may be too apt to forget the real world situation while exploring the math.  But I think it is true that some scientists are too willing to throw out the data that are just inconvenient. This is oversimplified in that everyone has their own approach and reasoning - hopefully with a mixture of theory and application, in both statistics and a specific field of science.  (I found trying to solve the real problems interesting, and that those with whom I worked who did not completely think of statistics as a 'black box' and would listen, appreciated the emphasis on relevance.) 

Finally, the previous comments by others above are enlightening, but one of them seems unique for a specific issue.  David wrote that "An econometric model asks for an answer to the question: What will Y be if I change X to be X + DX?"  That does seem to be an important distinction.

I hope that others will add more to this discussion, and/or add a related question to RG.  

Yes the coefficient and the multiples matters, but before going to the statistical models we have to think why we applying that and we have to interpret the results accordingly.