ROW_NUMBER()

This function is used to assign a unique id to each row returned by the sql query fired.

RANK()

This function is similar to ROW_NUMBER() with the only difference it leaves a gap between the groups i.e this function assign a distinct row id to distinct values, similar values have similar row id, the next row id is calculated as number of occurrence of previous row id + 1.   

Example:

Decalring a table variable and inserting values into it:

DECLARE @Table TABLE (
      Col_Value varchar(2)
)

INSERT INTO @Table (Col_Value)
      VALUES ('A'),('A'),('A'),('B'),('B'),('C'),('C');

Using ROW_NUMBER():

SELECT
      Col_Value,
      ROW_NUMBER() OVER (ORDER BY Col_Value) AS 'RowID'
FROM
      @Table;    

Output:

In above output a unique row id to each row returned.

Using RANK():

SELECT
      Col_Value,
      Rank() OVER (ORDER BY Col_Value) AS 'RowID'
FROM
      @Table;

Output:

In above output value ‘A’ is repeated thrice and has rank ‘1’, the next rank will be 1+3=4. Same with the next value 4+2=6 and so on