A central question of econometrics has been the question of causality: does X cause Y? Does the minimum wage cause employment? Do taxes cause capital flight?
Data science has been mostly concerned with the predictive power of its models: how can we sort pictures according to their contents? how can we sort customers according to their creditworthiness?
The two questions of causality and prediction are related: if we find a great predictor of customers' creditworthiness, does it mean that that characteristic causes creditworthiness? The answer is tricky, because excellent predictors are actually not always causal determinants of the outcome. If, for instance, high income students tend to cluster in good schools, this does not necessarily mean that household income causes student achievement. A perhaps more controversial example is that ethnicity predicts (i.e. is correlated with) a range of outcomes (education, health) but does not necessarily cause them.
Data science has become central to business applications (from Google Photos to face recognition in Facebook) but not so much to public policy analysis. In contrast, econometrics is ubiquitous in policy analysis.
There is some convergence though. A better, and more elaborate answer on how econometrics recently borrowed from data science is provided by Susan Athey (Sep 2016 podcast). http://www.econtalk.org/archives/2016/09/susan_athey_on.html