Machine learning is typically defined as a way to enable computers to learn from data to accomplish tasks, without explicitly telling them how.
Both fields can use logistic regression, regularization, and gradient descent to accomplish their goals, so in that sense there's no distinction.
But IMO there is a difference in their primary intention: econometrics typically focuses on inference about relationships, machine learning typically focuses on predictive accuracy. That's not to say that econometrics doesn't consider predictive accuracy, or that machine learning doesn't consider inference, but it's usually not their primary concern.
Logistic regression is clearly a classifier. And you need data to train it. So it's a supervised learning algorithm.
More generally, I recommend Breiman's two cultures article for some insight into the similarities and differences.
If you need a really simple explanation, then machine learning is a tool for generating predictions, while statistics is a method for performing inference about causes.