There is many difference in the gravity and layout-gravity. I am going to explain my experience about these 2 concepts(All information i got due to my observation and some websites).
Use Of Gravity and Layout-gravity in FrameLayout .....
Note:- 
- Gravity is used inside the View Content as some User have answer and it is same for all - ViewGroup Layout.
 
- Layout-gravityis used with the parent View as some User have answer.
 
- Gravity and Layout-gravityis work more useful with the- FrameLayoutchilds .- We can't use Gravity and Layout-gravityin FrameLayout's Tag ....
 
- We can set Child View any where in the - FrameLayoutusing- layout-gravity.
 
- We can use every single value of gravity inside the FrameLayout (eg:- - center_vertical,- center_horizontal,- center,- top, etc), but it is not possible with other ViewGroup Layouts .
 
- FrameLayoutfully working on- Layout-gravity.  Example:- If you work on- FrameLayoutthen you don't need to change whole Layout for adding new View. You just add View as last in the- FrameLayoutand give him- Layout-gravitywith value.(This is adavantages of layout-gravity with FrameLayout).
 
have look on example ...... 
<?xml version="1.0" encoding="utf-8"?>
<FrameLayout xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:layout_height="match_parent">
    <TextView
        android:layout_width="match_parent"
        android:layout_height="100dp"
        android:textSize="25dp"
        android:background="#000"
        android:textColor="#264bd1"
        android:gravity="center"
        android:layout_gravity="center"
        android:text="Center Layout Gravity"/>
    <TextView
        android:layout_width="wrap_content"
        android:layout_height="80dp"
        android:textSize="25dp"
        android:background="#000"
        android:textColor="#1b64b9"
        android:gravity="bottom"
        android:layout_gravity="bottom|center"
        android:text="Bottom Layout Gravity" />
    <TextView
        android:layout_width="wrap_content"
        android:layout_height="80dp"
        android:textSize="25dp"
        android:background="#000"
        android:textColor="#d75d1c"
        android:gravity="top"
        android:layout_gravity="top|center"
        android:text="Top Layout Gravity"/>
    <TextView
        android:layout_width="wrap_content"
        android:layout_height="80dp"
        android:textSize="25dp"
        android:background="#000"
        android:layout_marginTop="100dp"
        android:textColor="#d71f1c"
        android:gravity="top|right"
        android:layout_gravity="top|right"
        android:text="Top Right Layout Gravity"/>
    <TextView
        android:layout_width="wrap_content"
        android:layout_height="80dp"
        android:textSize="25dp"
        android:background="#000"
        android:layout_marginBottom="100dp"
        android:textColor="#d71cb2"
        android:layout_gravity="bottom"
        android:gravity="bottom"
        android:text="Top Left Layout Gravity"/>
</FrameLayout>
Output:-

Use Of Gravity and Layout-gravity in LinearLayout .....
Gravity working same as above but here differnce is that we can use Gravity inside the LinearLayout View and RelativeLayout View which is not possible in FrameLayout View.
LinearLayout with orientation vertical ....
Note:- Here we can set only 3 values of layout_gravity that is (left | right | center (also called center_horizontal)).
have look on example :- 
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:orientation="vertical"
    android:layout_height="match_parent">
    <TextView
        android:layout_width="wrap_content"
        android:layout_height="100dp"
        android:textSize="25dp"
        android:background="#000"
        android:textColor="#264bd1"
        android:gravity="center"
        android:layout_gravity="center_horizontal"
        android:text="Center Layout Gravity \nor \nCenter_Horizontal"/>
    <TextView
        android:layout_width="wrap_content"
        android:layout_height="80dp"
        android:textSize="25dp"
        android:background="#000"
        android:layout_marginTop="20dp"
        android:textColor="#d75d1c"
        android:layout_gravity="right"
        android:text="Right Layout Gravity"/>
    <TextView
        android:layout_width="wrap_content"
        android:layout_height="80dp"
        android:textSize="25dp"
        android:background="#000"
        android:layout_marginBottom="100dp"
        android:textColor="#d71cb2"
        android:layout_gravity="left"
        android:layout_marginTop="20dp"
        android:gravity="bottom"
        android:text="Left Layout Gravity"/>
</LinearLayout>
Output:-

LinearLayout with orientation horizontal ....
Note:- Here we can set also 3 values of layout_gravity that is (top | bottom | center (also called center_vertical)).
have look on example :-
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="match_parent"
    android:orientation="horizontal"
    android:layout_height="match_parent">
    <TextView
        android:layout_width="120dp"
        android:layout_height="100dp"
        android:textSize="25dp"
        android:background="#000"
        android:textColor="#264bd1"
        android:gravity="center"
        android:layout_gravity="bottom"
        android:text="Bottom \nLayout \nGravity"/>
    <TextView
        android:layout_width="120dp"
        android:layout_height="100dp"
        android:textSize="25dp"
        android:background="#000"
        android:layout_marginTop="20dp"
        android:textColor="#d75d1c"
        android:layout_gravity="center"
        android:text="Center \nLayout \nGravity"/>
    <TextView
        android:layout_width="150dp"
        android:layout_height="100dp"
        android:textSize="25dp"
        android:background="#000"
        android:layout_marginBottom="100dp"
        android:textColor="#d71cb2"
        android:layout_gravity="left"
        android:layout_marginTop="20dp"
        android:text="Left \nLayout \nGravity"/>
</LinearLayout>
output:- 

Note:- We can't use layout_gravity in the RelativeLayout Views but we can use gravity to set RelativeLayout childs to same position....