Regression coefficients vs feature_importances_ vs none
The main difference between Linear Regression and Tree-based methods is that Linear Regression is parametric: it can be writen with a mathematical closed expression depending on some parameters. Therefore, the coefficients are the parameters of the model, and should not be taken as any kind of importances unless the data is normalized. 

On the other hand, non-parametric methods have very different ways of measure the importance. In layman terms, measuring the importance of variables on a tree can be done by checking how close they appear to the root node. In ensembling methods, like bagging, one can compute the importance of a variable as the average among the ensemble, like in this stackoverflow answer.

The main difference is, then, the fact that parametric models have, through their parameters, a way of showing the importance of the variables, while non parametric models need some extra work.