By reading the excellent Statistical modeling: The two cultures (Breiman 2001), we can seize all the difference between traditional statistical models (e.g., linear regression) and machine learning algorithms (e.g., Bagging, Random Forest, Boosted trees...).
Breiman criticizes data models (parametric) because they are based on the assumption that the observations are generated by a known, formal model prescribed by the statistician, which may poorly emulate Nature. On the other hand, ML algos do not assume any formal model and directly learn the associations between input and output variables from the data.
I realized that Bagging/RF and Boosting, are also sort of parametric: for instance, ntree, mtry in RF, learning rate, bag fraction, tree complexity in Stochastic Gradient Boosted trees are all tuning parameters. We are also sort of estimating these parameters from the data since we're using the data to find optimal values of these parameters.
So what's the difference? Are RF and Boosted Trees parametric models?