I would agree with your assessment of things. ML is much more concerned with making predictions and a discipline like Econometrics, or Statisitcs, for instance, strives to find causation between variables.
ML excels at finding patterns in data and using these patterns for classification and prediction. Econometrics shares machine learners' interest in classification and prediction, as well as statisticians' concern with sample representativeness and sampling variance. As an aside, The discipline of statistics was born out of a desire to work with data efficiently, primarily by drawing relatively small samples from larger populations of interest instead of collecting data on everyone. As you know, people in the ML world will try to consume as much data as possible, whereas people in the statistics world, are taking samples of populations, with the understanding that a small subset is representative of an entire population, and that's good enough for the analysis that's being done.
Now, back to your question about proving causation. Correlation is a statistical technique which tells us how strongly the pair of variables are linearly related and change together. Causation takes a step further than correlation; it says any change in the value of one variable will cause a change in the value of another variable, which means one variable makes another happen. This is referred as cause and effect. Essentially, we can infer causality from a well designed randomized controlled experiment. Randomized and controlled don't intuitively belong together, but it's a complex dynamic. Think of the predator-prey model. As the number of prey increase, more predators can exist, but too many predators will decimate the prey population, so the number of predators will diminish, and then the number of prey will increase. This cycle continues over and over.
I did a quick Google search and came up with a couple of links, which seem to make a decent comparison between the two disciplines.
Hope that helps!!!