I emphasized here that both machine learning (ML) and econometrics (E) prominently feature prediction, one distinction being that ML tends to focus on non-causal prediction, whereas a significant part of E focuses on causal prediction. So they're both focused on prediction, but there's a non-causal vs. causal distinction. [Alternatively, as Dean Foster notes, you can think of both ML and E as focused on estimation, but with different estimands. ML tends to focus on estimating conditional expectations, whereas the causal part of E focuses on estimating partial derivatives.]
In any event, there's another key distinction between much of ML and Econmetrics/Statistics (E/S): E/S tends to be more concerned with probabilistic assessment of uncertainty. Whereas ML is often satisfied with point forecasts, E/S often wants interval, and ultimately density, forecasts.
There are at least two classes of reasons for the difference.
First, E/S recognizes that uncertainty is often of intrinsic economic interest. Think market risk, credit risk, counter-party risk, systemic risk, inflation risk, business cycle risk, etc.
Second, E/S is evidently uncomfortable with ML's implicit certainty-equivalence approach of simply plugging point forecasts into decision rules obtained under perfect foresight. Evidently the linear-quadratic-Gaussian world in which certainty equivalence holds resonates less than completely with E/S types. That sounds right to me. [By the way, see my earlier piece on optimal prediction under asymmetric loss.]