#Prolog, 143 bytes#
**Code:**

    q(X,N):-X<9,N=0;A is abs(X mod 10-X//10 mod 10),Y is X//10,q(Y,M),N is A+M*10.
    r(X,N):-X<9,N=X;q(X,Y),r(Y,M),N is X+M.
    p(X):-r(X,N),write(N).
**Explained:**

    q(X,N):-X<9,N=0;                                                         % If only one digit, the difference is 0
            A is abs(X mod 10-X//10 mod 10),Y is X//10,q(Y,M),N is A+M*10.   % Else, the difference is the difference between the last 2 digits + the recursive difference of the number without the last digit
    r(X,N):-X<9,N=X;                                                         % If we only have 1 digit the final answer is that digit
            q(X,Y),r(Y,M),N is X+M.                                          % Else, the final answer is the current number + the recursive difference of that number
    p(X):-r(X,N),write(N).         
**q** does the calculations that convert a number into it's Digit Difference.<br>
**r** recursively calls **q** and sums up the results to find the Digit Difference Sum.
<br>
**p** is the entry point. Takes a number, calls **r** and prints the answer.
<br><br>
**Example:**

    >p(8675309).
    8898683

Try it online [here](http://swish.swi-prolog.org/p/FEbBzrgH.pl).