When is the different in a number field a principal ideal? Q1: Do you known examples, where the different is not a principal ideal?
Q2: Is there a good interpretation for the reason, why this happens?
See e.g. Neukirch, Proposition 2.4, page 197.
The reason why I ask: the definition of the canonical additive character $\psi:x \mapsto \mathrm{e}^{2 \pi i (\mathrm{Tr}_{F / \mathbb{Q}} x \mod \mathbb{Z})}$ is somewhat unsatisfactory, if I want to identify the Pontryagin dual of the additive group of $\mathfrak{o}$ with the additive group of $F/\mathfrak{o}$, which simplifies some of notation involved when computing some $p$ adic integrals or Gauss sums.
Btw. with the canonical additive character, the Pontryagin duality is of the following form:
$$F / \mathfrak{D}^{-1} \cong \mathrm{Hom}_{ab.group} ( \mathfrak{o} , \mathbb{C}^\times).$$
where $\mathfrak{D}$ is the different and the isomorphism is given by
$$ \xi \in  F \mapsto \psi( \xi \cdotp).$$
 A: A partial answer:
Regarding Q1: An example for this is the number field generated by third root of $175$ .
See e.g. a comment by KConrad on this question
 Which number fields are monogenic? and related questions 
or also Ex 4.15 in these notes 
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/different.pdf
which contain several explicit examples related to differents.
Regarding Q2: I can give at least a good reason when it does not happen (that is when it is principal). Namely, when the ring of integers is generated by a single element. If it is generated by a single element $u$, then the different is generated by $f'(u)$ where $f$ is the minimal polynomial of $u$. 
A proof of this can be found in the above mentioned notes (Thm 4.3) as well as a discussion of the correct generalization of this result in the general case (Rem 4.5).  
Thus, for example, for quadratic fields it is always principal. And, more generally, one does not have to look for examples for non-principal in too 'nice' fields (i.e., those where the ring of integers is generated by one element, such as cyclotomic fields). 
