Is any difference between a MaterialButton and a simple Button?

If you are using a MaterialComponents Theme there is no difference between <Button /> and <com.google.android.material.button.MaterialButton />.

There is an auto-inflation enabled which will replace <Button with <com.google.android.material.button.MaterialButton at runtime.

The MaterialComponentsViewInflater replaces some framework widgets with Material Components ones at inflation time, provided if a MaterialComponents theme is in use.
Something similar happens also with AppCompat (you can check that MaterialComponentsViewInflater extends AppCompatViewInflater).

It means that, the <Button is replaced <com.google.android.material.button.MaterialButton at runtime, if you are using a MaterialComponents Theme.

If I want to get the behavior of the old plain button, I have to use:
<androidx.appcompat.widget.AppCompatButton />.

It is an option, but not necessarily.
It depends by what you want to achieve. You can also config a custom style since the Widget.MaterialComponents.Button style inherits by Widget.AppCompat.Button style.

It is the difference:

<style name="Widget.MaterialComponents.Button" parent="Widget.AppCompat.Button">
    <item name="enforceMaterialTheme">true</item>
    <item name="enforceTextAppearance">true</item>
    <item name="android:textAppearance">?attr/textAppearanceButton</item>
    <item name="android:textColor">@color/mtrl_btn_text_color_selector</item>
    <item name="android:paddingLeft">@dimen/mtrl_btn_padding_left</item>
    <item name="android:paddingRight">@dimen/mtrl_btn_padding_right</item>
    <item name="android:paddingTop">@dimen/mtrl_btn_padding_top</item>
    <item name="android:paddingBottom">@dimen/mtrl_btn_padding_bottom</item>
    <item name="android:insetLeft">0dp</item>
    <item name="android:insetRight">0dp</item>
    <item name="android:insetTop">@dimen/mtrl_btn_inset</item>
    <item name="android:insetBottom">@dimen/mtrl_btn_inset</item>
    <item name="android:stateListAnimator" ns2:ignore="NewApi">@animator/mtrl_btn_state_list_anim</item>
    <item name="cornerRadius">@null</item>
    <item name="elevation">@dimen/mtrl_btn_elevation</item>
    <item name="iconPadding">@dimen/mtrl_btn_icon_padding</item>
    <item name="iconTint">@color/mtrl_btn_text_color_selector</item>
    <item name="rippleColor">@color/mtrl_btn_ripple_color</item>
    <item name="backgroundTint">@color/mtrl_btn_bg_color_selector</item>
    <item name="shapeAppearance">?attr/shapeAppearanceSmallComponent</item>
  </style>