Why is u-equivalence useful? If $X$ is a standard set, we define, for $\alpha\in\ ^*X$,
$U_{\alpha}=\{A\subset X\mid\alpha\in\ ^*A\}$. We can see that $U_\alpha$ is an ultrafilter on $X$.
We thus define an equivalence relation on $^*X$ which is $\alpha\sim\beta$ iff $U_{\alpha}=U_{\beta}$.
Is it possible, using big enough saturation, that there are distinct $\alpha,\beta\in\ ^*X$ such that $\alpha\sim\beta$ (for example, using this) ?

The paper by Di Nasso you cite was published in 2002.  In a latter paper by Benci, Forti and Di Nasso:

Benci, Vieri; Forti, Marco; Di Nasso, Mauro.
  The eightfold path to nonstandard analysis. Nonstandard methods and applications in mathematics,  3–44, 
  Lect. Notes Log., 25, Assoc. Symbol. Logic, La Jolla, CA, 2006

in section 9 seeks to present models of hyperintegers as subsets of the Stone-Cech compactification of $\mathbb Z$.  This suggests that for those models distinct integers necessarily correspond to distinct ultrafilters. 
Meanwhile Di Nasso in chapter 11 of Nonstandard analysis for the working mathematician, page 445, points out that $c^+$ saturation implies that the map from hypernaturals to $\beta \mathbb N$ (the Stone-Cech compactification) is onto. It suffices to choose a model of the hypernaturals of high enough cardinality (namely, higher than that of $\beta\Bbb N$) to ensure that there are distinct hypernaturals that define identical ultrafilters on $\Bbb N$.
