Understand that using this example 
android:layout_gravity
- android:layout_gravityis used to set the position of an element in
its parent (e.g. a child- Viewinside a- Layout).
- Supported by LinearLayout and FrameLayout
android:gravity
- android:gravityis used to set the position of content inside an
element (e.g. a text inside a- TextView).
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:paddingBottom="@dimen/activity_vertical_margin"
    android:paddingLeft="@dimen/activity_horizontal_margin"
    android:paddingRight="@dimen/activity_horizontal_margin"
    android:paddingTop="@dimen/activity_vertical_margin"
    android:orientation="vertical">
    <LinearLayout
        android:layout_width="wrap_content"
        android:layout_height="0dp"
        android:layout_weight="1"
        android:orientation="vertical"
        android:layout_gravity="left"
        android:gravity="center_vertical">
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/first"
            android:background="@color/colorPrimary"
            android:gravity="left"/>
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/second"
            android:background="@color/colorPrimary"
            android:gravity="center"/>
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/third"
            android:background="@color/colorPrimary"
            android:gravity="right"/>
    </LinearLayout>
    <LinearLayout
        android:layout_width="wrap_content"
        android:layout_height="0dp"
        android:layout_weight="1"
        android:orientation="vertical"
        android:layout_gravity="center"
        android:gravity="center_vertical">
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/first"
            android:background="@color/colorAccent"
            android:gravity="left"/>
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/second"
            android:background="@color/colorAccent"
            android:gravity="center"/>
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/third"
            android:background="@color/colorAccent"
            android:gravity="right"/>
    </LinearLayout>
    <LinearLayout
        android:layout_width="wrap_content"
        android:layout_height="0dp"
        android:layout_weight="1"
        android:orientation="vertical"
        android:layout_gravity="right"
        android:gravity="center_vertical">
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/first"
            android:background="@color/colorPrimaryDark"
            android:gravity="left"/>
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/second"
            android:background="@color/colorPrimaryDark"
            android:gravity="center"/>
        <TextView
            android:layout_width="@dimen/fixed"
            android:layout_height="wrap_content"
            android:text="@string/third"
            android:background="@color/colorPrimaryDark"
            android:gravity="right"/>
    </LinearLayout>
</LinearLayout>
Which gets rendered as following:

Now you see 3 colors here thats because of the textView background,
lets take android:layout_gravity,here its used in LinearLayouts, they are positioned inside their parent , left in the parent , middle in the parent , right in the parent (see LinearLayout has/can have children)
lets take android:gravity check those textViews each color has 3 of them and see how they(their content) are possition inside their element (does a text view has children? or only a content?)
I feel this like
gravity - arrange content inside the view , view does not want to move but the content move
layout_gravity - Its not about the content it's the layout which move here and there
Go through the original question as well its cool