Differential privacy definition Differential privacy defines "privacy" of a mechanism $A$ as the "closeness" of the two distribution $Pr[A(D) \in S]$ and $Pr[A(D') \in S]$ where $D,D'$ differ in one element. And the distance between these distributions is multiplicative, i.e. 

$\left| \frac{Pr[A(D) \in S]}{Pr[A(D') \in S]} \right| \leq e^\epsilon$

I have difficulty in understanding this choice of this multiplicative distance measure, as opposed to the standard distance (statistical difference) as in cryptography (indistinguishability), i.e. 

$|Pr[A(D) \in S] - Pr[A(D') \in S]| \leq neg(.)$

The paper "Calibrating Noise to Sensitivity in Private Data Analysis" by Dwork et al. put forth two reasons for using the multiplicative distance: 


It is more stringent, since if one probability is 0, the other must also be 0 (which is not guaranteed when using standard measure of statistical difference). This, I understand. 
The leakage (distance between the distributions) must be non-negligible for the utility of the mechanism. This I really struggle to understand. Could anyone give an simple example to illustrate this, please?
I can't give an example illustrating why leakage must be non-negligible for the utility of the mechanism, but I can give a proof of why leakage must be non-negligible for the utility of the mechanism.



Let $\;\; U \: = \: \left\{\langle D,D'\rangle : D,D' \text{ differ in one element}\right\} \;\;\;$.

By the triangle inequality, for all $D$ and $D'$, for all $S$,

$|\operatorname{Pr}(A(D) \in S)-\operatorname{Pr}(A(D') \in S)|$
$\leq$
number_of_elements $\cdot \operatorname{sup}(\{|\operatorname{Pr}(A(D) \in S)-\operatorname{Pr}(A(D') \in S)| : \langle D,D'\rangle \in U\}$

If number_of_elements is polynomial and $\:\operatorname{sup}(\{|\operatorname{Pr}(A(D) \in S)-\operatorname{Pr}(A(D') \in S)| : \langle D,D'\rangle \in U\})\:$ is negligible, then

for all $D$ and $D'$, for all $S$, $\:|\operatorname{Pr}(A(D) \in S)-\operatorname{Pr}(A(D') \in S)|\:$ is negligible.

If number_of_elements is polynomial and

$\operatorname{sup}(\{|\operatorname{Pr}(A(D) \in S)-\operatorname{Pr}(A(D') \in S)| : \langle D,D'\rangle \in U\})$

is negligible, then $A$ is statistically close to being independent of $D$.

Therefore, if $\:\operatorname{sup}(\{|\operatorname{Pr}(A(D) \in S)-\operatorname{Pr}(A(D') \in S)| : \langle D,D'\rangle \in U\})$

is negligible then the mechanism is not useful.