Why do the characters not determine the group?

One possible answer is given by the following paper.

Hoehnke, H.-J.; Johnson, K. W. The 1-, 2-, and 3-characters determine a group. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 243–245.

Here is a brief explanation. For a character $\chi$ of a finite group $G$, define the corresponding $2$-character $\chi^{(2)} : G \times G \rightarrow \mathbb{C}$ by $$\chi^{(2)}(x,y) = \chi(x)\chi(y) - \chi(xy)$$ for all $x, y \in G$. We also define the corresponding $3$-character $\chi^{(3)}: G \times G \times G \rightarrow \mathbb{C}$ by $$\chi^{(3)}(x,y,z)=\chi(x)\chi(y)\chi(z)−\chi(x)\chi(yz)−\chi(y)\chi(xz)−\chi(z)\chi(xy)+\chi(xyz)+\chi(xzy)$$ for all $x, y, z \in G$.

There are similar definitions of $k$-characters $\chi^{(k)}$ and these go back to Frobenius. A paper of Formanek and Sibley from 1991 shows that the group determinant determines $G$. One consequence of this is that $G$ is determined by its irreducible characters $\chi$ and their $k$-characters $\chi^{(k)}$.

In their paper, Hoehnke and Johnson improved this by showing that the irreducible characters $\chi$ along with $\chi^{(2)}$ and $\chi^{(3)}$ suffice to determine $G$:

Theorem. Let $G$ be a finite group with complex irreducible characters $\chi_1$, $\ldots$, $\chi_t$. Then $G$ is determined up to isomorphism by the $\chi_i$, $\chi_i^{(2)}$, $\chi_i^{(3)}$, $1 \leq i \leq t$.

In a sense this result is optimal, since in the following paper Johnson and Sehgal show that the knowledge of $\chi$ and $\chi^{(2)}$ does not determine $G$ in general.

Johnson, Kenneth W.; Sehgal, Surinder K. The 2-character table does not determine a group. Proc. Amer. Math. Soc. 119 (1993), no. 4, 1021–1027.

The following papers might be of interest

1. Sandro Mattarei. Character tables and metabelian groups. J. London Math. Soc. (2) 46 (1992), no. 1, 92-100.
2. Sandro Mattarei. An example of $p$-groups with identical character tables and different derived lengths. Arch. Math. (Basel) 62 (1994), no. 1, 12-20.
3. Sandro Mattarei. On character tables of wreath products. J. Algebra 175 (1995), no. 1, 157-178.