On finite groups with same complex-valued character table

Finite groups have the same complex character tables if and only if their group algebras are isomorphic as quasi-Hopf algebras (if and only if the group algebras are twisted forms of each other as Drinfel'd quasi-bialgebras, if and only if there is non-associative bi-Galois algebra over these groups). For details see arXiv:math/0001119.

This is a complicated question. A pair of non-isomorphic groups with the same character table is sometimes called a "Brauer Pair". There are many such pairs, especially among $p$-groups.