The set of imputations may be empty, while the set of preimputations is never empty. Hence the nucleolus may be empty, while the prenucleolus is never empty. This happens, for example, in the two-player game v(1)=v(2)=0 and v(1,2)=-1.
If you look for a game where both nucleolus and prenucleolus are nonempty and yet they differ, Exercise 20.15 in "Game Theory" by Maschler-Solan-Zamir asks to compute the nucleolus and the prenucleolus of the three-player game where v(1,2)=1 and v(S)=0 for every other coalition S. The nucleolus is (0,0,0), since this is the only imputation. I do not recall what is the prenucleolus, but my guess is that it is (1/4,1/4,-1/2), where the maximal excess is 1/2.

The sequel is an edit that includes my other clarification:

In the prenucleolus, symmetric players obtain the same payoff, hence the prenucleolus has the form $(x,x,-2x)$ for some real number $x$.
The prenucleolus is the preimputation that minimizes the maximal excesses (in lexicographic order). Let us start by finding the $x$ that minimizes the maximal excess $max_S E(S,x)$. 
For each coalition $S$ draw the graph $x \mapsto E(S,x) = v(S) - x(S)$. This graph appears below. The maximal excess for a given $x$ is the upper contour $\max\{ 1-2x,2x\}$, whose minimum is attained at $x=1/4$. Hence the prenucleolus is $(\frac{1}{4},\frac{1}{4},-\frac{1}{2})$.
[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/CCcDA.jpg