When to use MaterialToolbar and androidx Toolbar and the difference between both?

The biggest difference is the use of a MaterialShapeDrawable as background implementing some Material features, such as shaping and elevation overlays for Dark Themes.

You can find the answer in the docs of MaterialToolbar: https://developer.android.com/reference/com/google/android/material/appbar/MaterialToolbar

Simply MaterialToolbar extends androidx.appcompat.widget.Toolbar and let it behave as MaterialComponent. The main feature seems to be elevation. You can read through the implementation here:

https://github.com/material-components/material-components-android/blob/master/lib/java/com/google/android/material/appbar/MaterialToolbar.java